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Modelling flow-induced vibrations of gates in hydraulic structures
Erdbrink, C.D.
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Citation for published version (APA):Erdbrink, C. D. (2014). Modelling flow-induced vibrations of gates in hydraulic structures
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Download date: 26 Jun 2018
Modelling flow-induced vibrations of gates in hydraulic structures
Modelling flow
-induced vibrations of gates in hydraulic structuresChristiaan D
. Erdbrink
Christiaan D. ErdbrinkISBN 978-90-6464-800-7
MODELLING FLOW-INDUCED VIBRATIONS
OF GATES IN HYDRAULIC STRUCTURES
MODELLING FLOW-INDUCED VIBRATIONS
OF GATES IN HYDRAULIC STRUCTURES
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Universiteit van Amsterdam
op gezag van de Rector Magnificus
prof. dr. D. C. van den Boom
ten overstaan van een door het college voor promoties ingestelde
commissie, in het openbaar te verdedigen in de Agnietenkapel
op woensdag 3 september 2014, te 14:00 uur
door
Christiaan Dirk Erdbrink
geboren te Leidschendam
Promotor: Prof. dr. P.M.A. Sloot Copromotor: Dr. V.V. Krzhizhanovskaya Overige leden: Prof. dr. A.V. Boukhanovski Prof. dr. A.E. Eiben Prof. dr. ir. S.N. Jonkman Prof. dr. R.J. Meijer Prof. dr. R.P. Stevenson Faculteit: Faculteit der Natuurwetenschappen, Wiskunde en Informatica The work presented in this dissertation has been carried out at the Section of Computational Science of the University of Amsterdam, The Netherlands and at the Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, Russian Federation. Christiaan acknowledges financial support by:
Deltares (www.deltares.nl),
Leading Scientist Program of the Russian Federation, contract 11.G34.31.0019,
“5-100-2020” Program of the Russian Federation, grant 074-U01.
Author contact: [email protected] Published and printed by GVO drukkers & vormgevers B.V. | Ponsen & Looijen ISBN: 978-90-6464-800-7
Computational Science
Contents1 Introduction...........................................................................................................................................................7
1.1 Motivation..................................................................................................................................................7
1.2 Terminologyofcomputationalmodelling.................................................................................10
1.2.1 Modelbuilding.................................................................................................................................10
1.2.2 Modeluse............................................................................................................................................12
1.3 Thesisaims..............................................................................................................................................14
1.4 Thesisset‐up..........................................................................................................................................15
2 Background.........................................................................................................................................................17
2.1 Introduction............................................................................................................................................17
2.2 Hydrodynamics:flowaroundastaticgate...............................................................................17
2.2.1 Generaldescription........................................................................................................................17
2.2.2 Turbulence.........................................................................................................................................20
2.2.3 Hydrodynamicforcesonastaticunderflowgate.............................................................21
2.2.4 Navier‐Stokesequations..............................................................................................................24
2.3 Linearvibrations..................................................................................................................................25
2.3.1 Basictheory.......................................................................................................................................25
2.3.2 Frequencydomain..........................................................................................................................28
2.4 Physicsofflow‐inducedgatevibrations....................................................................................29
2.4.1 Introductiontoflow‐inducedvibrations..............................................................................29
2.4.2 Dimensionlessparameters.........................................................................................................30
2.4.3 Causesofflow‐inducedgatevibrations................................................................................32
2.4.4 Addedcoefficients..........................................................................................................................36
2.4.5 Consequencesofgatevibrations..............................................................................................38
2.4.6 Mythsaboutgatevibrations......................................................................................................39
2.5 Generalisationandproblemsolvinginpractice.....................................................................40
2.6 Conclusionsfromthephysicaldescription...............................................................................42
3 Physics‐basednumericalmodelling........................................................................................................45
3.1 Introduction............................................................................................................................................45
3.2 ApplyingCFDforproblemsolvinginhydraulicengineering...........................................45
3.3 SettingupaFIVmodel.......................................................................................................................49
3.3.1 Finiteelementmethod.................................................................................................................49
3.3.2 Turbulencemodelling...................................................................................................................49
3.3.3 ThearbitraryLagrangian‐Eulerianmesh............................................................................50
4 Multi‐scalemodelfordischargecontrolandflowimpact.............................................................55
4.1 Introduction............................................................................................................................................55
4.2 Approachandmethod........................................................................................................................56
4.2.1 General.................................................................................................................................................56
4.2.2 Configurationsofmulti‐gatedstructure...............................................................................57
4.2.3 Systemmodelandgatecontrol................................................................................................58
4.2.4 Dischargemodel..............................................................................................................................61
4.3 CFDsimulations....................................................................................................................................64
4.3.1 Modelset‐up......................................................................................................................................64
4.3.2 Analysisofsimulationoutput:flowimpact.........................................................................66
4.4 Modelvalidation...................................................................................................................................67
4.5 Resultsoftestcasesimulations.....................................................................................................69
4.5.1 Resultsofsystemanddischargemodel................................................................................69
4.5.2 ResultsofCFDsimulations.........................................................................................................72
4.5.3 Resultsofflowanalysis................................................................................................................74
4.6 Modelcouplingtests...........................................................................................................................76
4.7 Conclusionsofthischapter..............................................................................................................78
5 Physicalexperiment........................................................................................................................................79
5.1 Preface.......................................................................................................................................................79
5.2 Introduction............................................................................................................................................79
5.3 Experimentalset‐up............................................................................................................................81
5.4 Definitions...............................................................................................................................................86
5.5 Measurementconditionsandvariationofparameters.......................................................87
5.6 Resultsofphysicalexperiment......................................................................................................88
5.7 Comparisonwithotherexperimentalresults..........................................................................91
5.8 Summary..................................................................................................................................................91
5.9 Photographsfromtheexperiment...............................................................................................91
6 Numericalsimulationstudyofgatevibration.....................................................................................97
6.1 Introduction............................................................................................................................................97
6.2 Modelset‐up...........................................................................................................................................97
6.3 Selectedcasesandmodelvalidation........................................................................................100
6.3.1 Addedmassvalidation..............................................................................................................100
6.3.2 Artificialaddedcoefficients.....................................................................................................101
6.4 Resultsofcalibratednumericalmodel....................................................................................102
6.5 CombineddiscussionofphysicalexperimentandFEMmodelling.............................116
6.5.1 Vibrationmechanism.................................................................................................................116
6.5.2 Effectofflowthroughtheventilatedgate.........................................................................117
6.5.3 Implementationinprototypegates.....................................................................................118
6.5.4 Evaluationofnumericalmodelling......................................................................................118
6.6 CombinedconclusionsofphysicalexperimentandFEMmodelling..........................119
6.7 Summary...............................................................................................................................................120
6.8 Additionalsimulations:flowvelocityandpressureofcase1.......................................121
6.9 Additionalsimulations:analternativeventilationdesign..............................................122
7 Data‐drivenoperation................................................................................................................................125
7.1 Introduction.........................................................................................................................................125
7.2 Whatquantitiestomeasure?.......................................................................................................126
7.3 Systemset‐up......................................................................................................................................127
7.3.1 Overviewofthecomponents..................................................................................................127
7.3.2 Machinelearningmodule.........................................................................................................128
7.3.3 Physics‐basedmodel..................................................................................................................131
7.4 Resultsofexperimentaldataclassification...........................................................................131
7.5 Applicationchallenges....................................................................................................................133
7.5.1 General..............................................................................................................................................133
7.5.2 Multipledegreesoffreedom...................................................................................................134
7.5.3 Applyingmachinelearninginengineering......................................................................135
7.6 Conclusions..........................................................................................................................................136
8 Evolutionarycomputingforsystemidentification.......................................................................137
8.1 Introduction.........................................................................................................................................137
8.2 Meta‐heuristicsandevolutionarycomputing......................................................................139
8.3 Differentialevolution.......................................................................................................................140
8.4 Identifyingself‐excitedvibrationsusingdifferentialevolution...................................142
8.4.1 Approach..........................................................................................................................................142
8.4.2 Results...............................................................................................................................................143
8.4.3 Sensitivityanalysis......................................................................................................................147
8.4.4 ImprovingthesearchbyincludingresultsfromFFT..................................................149
8.4.5 Discussion.......................................................................................................................................152
8.5 Applicationofevolutionarysystemidentificationtoexperimentaldata.................153
8.6 Geneticprogrammingandsymbolicregression..................................................................156
8.7 Conclusionsandoutlook................................................................................................................157
9 Inferringnumericalalgorithms..............................................................................................................159
9.1 Introduction.........................................................................................................................................159
9.2 Inferringsolveralgorithms...........................................................................................................160
9.2.1 Introduction...................................................................................................................................160
9.2.2 Background.....................................................................................................................................162
9.2.3 Methodforgeneratingcomputationalschemesbyanevolutionaryalgorithm164
9.2.4 Resultsanddiscussion...............................................................................................................166
9.3 Conclusions..........................................................................................................................................172
9.4 ReflectionontheworkinChapters8and9..........................................................................172
10 Conclusionsandperspectives...........................................................................................................175
10.1 Collectedconclusions......................................................................................................................175
10.2 Reflection..............................................................................................................................................176
10.3 Perspectivesandrecommendations.........................................................................................177
References..................................................................................................................................................................179
Publications...............................................................................................................................................................189
Summary.....................................................................................................................................................................191
Samenvatting............................................................................................................................................................193
Postscript....................................................................................................................................................................195
Curriculumvitae......................................................................................................................................................198
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1 Introduction
1.1 Motivation
Flooddefencesystemsareofparamount importance for thesafetyofvulnerable low‐lyingareas. In coastal and riverine regions civil engineering structures are built to control thewaterlevelsandtoprotectthehinterland.TheriverdeltaofTheNetherlandsisanexampleofadenselypopulatedandeconomicallysignificantareawheremanyhydraulicstructureswereconstructedforthispurpose.Gatesaremovableelementsofbarriersand formessentialpartsof flooddefencesystems,becausetheyregulatethedischargebetweenbodiesofwater.Managementofinlandwaterlevelsisunthinkablewithoutgatedweirsinriversandchannels.Coastalsluicestructuresusegatestoprovideoutflowtotheseaandprotectionfromhighsealevelsatthesametime.Inbusynavigationchannels large floatingsectorgatesarestandbytobeclosedduringstormsurge events. These structures are for instance found near Rotterdam and in SaintPetersburg, Russia, where a large damwithmultiple gate sections protects the city fromfloods. Smaller gates are found in pipes and in culverts of navigation locks. Reliableoperationofgatesofcivilengineeringstructuresandpreventionofgate failurearecrucialforthereliabilityandsafetyofflooddefences.Hydraulicengineeringisthebranchofcivilengineeringthatisconcernedwiththedesignofstructures related to the flow and transportation of fluids: hydraulic structures. This fieldcombinesstructuralengineeringwithhydrodynamics.Thegatesofhydraulicstructuresarecalled hydraulic structure gates. The adjective ‘hydraulic’ in this context can always betraced back to the hydraulic engineering discipline and should not be confused withhydraulicdevicesusedtooperateallsortsofmechanicalappliances,suchas–indeed–gates.Hydraulic structure gates are found inmany shapes and sizes as part of various types ofstructures. Their primary function is in all cases the regulation of flow discharge, waterpressureorwaterlevels.Dependingontheapplication,gatesinhydraulicstructuresmaybecompletely opened, closed, or partly opened for long periods of time. This defines thechallengeofdesignandoperation:theyneedtohavethestrengthtoendureextremeforces,andtheymustbeopenedorclosedatexactlytherightmoment.Thefactthathydraulicgatesareindirectcontactwithpowerfulflowsmakesthejudgementandquantificationofforcesonthegatesdemanding.Theinteractionofhydraulicgateswithpassingcurrentscanleadtoflow‐inducedvibrations(FIV).Undercertaincircumstancesthesevibrations can grow in size and attain the form of regular oscillations with considerableamplitudes. Such manifestation of FIV poses a threat to hydraulic gate operation. Theassociated exposure to impermissible dynamic forces can lead to unexpected downtime,untimelymaintenance,andevenfailureduringemergencysituations. Inextremecases,thestabilityofthestructureasawholeisthreatened.
8
Thedesignofgatesofhydraulicstructureselementarilyfollowsthesameapproachasotherstructuralelements.There isacompromisebetweenseveral criteriadictatedby thegate’schieffunction.Inaproperdesignprocedure,staticanddynamicloadingcasesareidentifiedandassessed.Forstaticdesign forces,analyticalapproachesprovidereasonableestimates;rulesof thumbbasedonextremestatic loading situations e.g.maximumheaddifference facilitateengineeringdecisions.Fordynamicloadingthisformofquantificationisusuallynotanoption.ThereasonwhyFIVofgatesdeferastraightforwardanalyticalapproach is thatthe combination of structural details and flow conditions leads to complex interplays, thedetailsofwhicharedecisivefortheinitiationandgrowthofvibrationsandrelateddynamicforces.Everystudyinhydraulicengineering–foracademicandconsultancypurposesalike–startswithachoicebetweenphysicalmodelling,numericalmodellingandadeskstudybasedonliteratureandexpertknowledge.Physicalmodellingcanbea laboratorystudyofascaled‐downversionofa partofa structureorfieldmeasurementsoftheactualstructure,calledprototypeorinsitumeasurements.Themostprominentandtraditionalwayofinvestigatinggate vibrations, as reflected in publication numbers, is through physical scale modelexperiments in the laboratory supplementedwithanalytical studies for achievingphysicalinterpretationsofthemeasurementdata e.g.Kolkman,1984;ThangandNaudascher,1986;Jongeling,1988 .Thebulkofexperimentalresearchwasdoneintheperiod1965‐1995,withonlyasmallnumberofstudiesat laterdates e.g.BilleterandStaubli,2000 .Experimentalstudies performed in The Netherlands were partly motivated by the construction ofhydraulicstructuresfortheDeltaworksproject,whichendedin1997withtheconstructionoftheMaeslantbarrier,seeFigure1.1.TheDutchlaboratoryresearchongatevibrationswasdone at the research instituteDeltares formerlyWL|DelftHydraulics . Knowledge on thedynamicbehaviourofhydraulicgatesgainedbyphysicalscalemodelstudiesandfieldtestsiscollectedinKolkmanandJongeling 1996 .
Figure 1.1. TheMaeslant barrier in The Netherlands, a storm surge barrier with two floating gates(picturefromhttps://beeldbank.rws.nl,Rijkswaterstaat).
9
Pastphysicalmodelresearchhasproducedareasonablygoodunderstandingofthephysicalworking of gate vibrations (see Section 2.3). Design manuals nowadays contain shortoverviewswithbeneficialgategeometriesandwarnforundesiredconstructiondetails(e.g.Novak et al., 2007). In spite of this, problems still arise in practice with unpredictablefrequency.Thisisattributedtoone,oracombinationof,thefollowingcauses:‐Thegatedesigncontainssuboptimal characteristicswith respect toFIVbecause theproblemwasunderestimatedoroverlookedcompletely.
‐Physicalmodeltestsweredonetocheckandimprovedynamicresponsepropertiesofthe gate section, but unfavourable conditions nevertheless led to vibrations (e.g.Thang,1990).Despiteawarenessinthedesignstage,thereisstillariskofunknownorunanticipatedfailuremodesoccurringinreallife.
‐Negligenceorignoranceleadingtoimproperoperationaldecisions.‐Achangeofhydraulicconditionsduringthelifetimeofthestructuresuchthatdesignconditionsareexceeded.
‐ New operating procedures are introduced, for example in connectionwith updatedenvironmentalrequirements.
‐Gradualdeteriorationofthestructureorsubstandardmaintenancecancontributetoadifferentdynamicresponse.
This list reflects the fact that gate vibrations constitute a non‐standard point of attentionwithin design and operation of hydraulic structures. Preventive measures are as a rulerelativelyeasytoimplementwhentheissueisrecognisedatanearlystage(i.e.wellbeforefinalisingthedesign),butconsequencescanbecomefar‐reachinglateron.The tremendous increase in available computing power in the last three decades haspromoted the usage of computational fluid dynamics (CFD) in environmental engineeringprojects(Batesetal.,2005).CFDmodelshavebecomehouseholdtoolsforsolvingproblemsinhydrology(SolomatineandOstfeld,2008),coastalmanagement(Roelvinketal.,2009)andriver morphology (Van Rijn 1987, Mosselman and Sloff, 2008). Moreover, they are beingusedinoperationalsystemsinthoseareas.Veryfewexamplesofmodelapplicationsexistinwhich vibrations of barrier gates are simulated, however. Numerical research on FIV andfluid‐structureinteraction(FSI)hasfocusedonmoreprominentfieldsofapplication,suchasaerospace engineering or nuclear power engineering, and fundamental activities, such asbenchmarkingofcomputationalmodels.Asaresult,therearenooff‐the‐shelfcomputationalmodels today that are capable of simulating hydraulic gate dynamics to an extent thatenablespracticalusage.Investigations into the reduction of gate vibrations for the large Saint Petersburg barrierFigure1.8 involvedbothphysicalandnumericalmodelling,seeKlimovichetal. 2006 andLupuleacet al. 2008 .This resulted in anadjusteddesignof thebottompartof themainsector gates. Construction of this barrier finished in 2010. For the large barriers in TheNetherlands,noCFDanalyseswereperformedtosimulateandassesstheFIVloads.FortheMaeslant barrier hydraulic design conditions were determined and physical scale modeltests of the structure design were done specifically for checking FIV properties; designimprovementsweremadeaccordingly.
10
Themotivationforthisthesisfirstofalloriginatesfromfirst‐handexperienceinprojectsatDeltaresintheperiod2009‐2011involvingreal‐lifestructures,amongwhichtheHaringvlietdischargesluices (Figure1.2).Prior toengaging in thePhD,physicalmodelmeasurementswere done in the laboratory of Deltares, reported in Erdbrink (2012). An underlyingmotivation is also to raise the level of knowledgeonhydraulic gates andgenerally on thedynamics of hydraulic structures –knowing that this remains a salient topic indesign andmaintenanceoflargestructuresworldwide–andtoimprovejudgementofvibrationrisks.
Figure1.2.DischargethroughagateoftheHaringvlietbarrier,TheNetherlands(picturebyDeltares).
1.2 Terminologyofcomputationalmodelling
Because computational models are employed in a variety of ways in this thesis, it isimportant tomake clear from the start how they are built and used. Terminology can bequitedifferentacrossresearchfieldsandthereforeazoomed‐outviewisneeded.Strangely,the distinction between model building and model usage is not always recognised. Thisomissionisasignthataproperviewontherelationbetweenproblemdescriptionandmodelemployment could be lacking, notwithstanding the fact that sometimes for a computerscientistmakingamodel is theultimategoal–and,sometimes, foranengineerapplyingamodelistheonlythingthatmatters.1.2.1 ModelbuildingThe traditional approach to setting up a numerical model the adjectives numerical andcomputational are used interchangeably is from a mathematical model, which is adescriptionof thephysicalprocess in formulae.Thismathematicalmodel isa resultof theevaluationofmeasureddataofthephysicalprocessbyanexpert.SeeFigure1.3.Oftentheevaluation that is used to define the mathematical model is not explicitly based on new
11
measurements,orisskippedaltogether,whenthemathematicalequationsareadoptedfrompreviousstudies.
Figure1.3.Buildinganumericalmodelfromamathematicalmodel.Inthisapproachthemodelstructureconsistsofmathematicalexpressionsthatarederivedfromananalyticalendeavourtocapturethephysicalprocess.Themagicofthemathematicalmodel here is that it acts as a bridge: it provides the numerical relations necessary forcomputationanditcontainscompactphysicaldescriptionsthataremeaningfuloutsideofthenumericalrealm.SeeStellingandBooij(1999)foraguidelinehowthisapproachisusedinhydraulicengineeringtoestablishaconnectionbetweennumericalmodelsandreality.A second, different approach is called system identification (SI), defined in Figure 1.4. Ameasured time signal u acts simultaneously as input for the process (or system) beingstudiedandasinputforthenumericalmodel.Thedifferenceoftheoutputsyand ,theerrorsignal,issubsequentlyusedtoadaptthenumericalmodel;thisimprovesitsperformance.Inmostreal‐lifesituationsthemeasuredprocessoutputisinfluencedbynoise.
Figure1.4.Settingupanumericalmodelbysystemidentification.AfterNelles(2001).Theactofrepeatedlyapplyingtheerrorsignalforimprovementofthemodel i.e.ofreducing| | is called training. Inmachine learning, originally abranchof artificial intelligencebuttodayarguablytooextensiveatopic itself tobeconsideredasingleresearchfield, thistypeoftrainingiscalledsupervisedlearning.Thismeansthatbothinputandoutputsignalsare available for gradually improving the model. Note that these ‘signals’ can have allimaginable shapes: they can contain a physical quantity varying in time, for examplepressurep t ,buttheycanalsobeadiscretevariablexthatindicatesinclusionorexclusionof an object in a set, for example x ∊ blue, red . The model resulting from the trainingbasically gives well‐tweaked rules for a mapping from input to output. The described
processevaluationu
noise
ymathematical
modelnumerical
model
process
numerical model
u error
noise
y
y
12
approachmakesintensiveuseofthe measured dataandisthereforetermedempiricalordata‐drivenmodelling.Thefirstdescribedapproach(Figure1.3)basedonamathematicalmodelthatdescribesthefundamental principles is referred to as physics‐basedmodelling and providesmaximuminsightsintheprocess.Itisaformofwhite‐boxmodelling.Thesecondapproach(Figure1.4)isanexampleofblackboxmodelling.Thissecondapproachrequiresaminimumofdomainexpertiseandcangiveaptnumericalrepresentationsofcomplicatedsystemsthatarebadlyunderstood.Onthedownside,itsaccuracydependsonthe(qualityof)theavailabledataandextrapolation to rangesoutsideof trainingconditions isprecarious.Needless tosay,manyhybridformsweredevelopedthataresometimescalledgreymodels.1.2.2 ModeluseDepending on the problem type that is to be solved, there are various ways to use thenumericalmodel.Figures5to7schematicallyshowhowcomputationalmodelscanbeused(afterNelles,2001).Inthefigures,u(t)isaninputvalueattimetandy(t)and (t)areoutputvalues fromtheprocessand themodel, respectively.Predictivemodelsusepast inputandoutputsignalsfromamonitoredprocesstocomputeapredictionforoneormorestepsintothefuture.IntheleftschemeofFigure1.5,aone‐stepaheadpredictionisshown.
Figure1.5.Useofnumericalmodels.Left:prediction.Right:simulation.Simulationisdifferent.Here,thenumericalmodelisusedtomimictheprocessoutputusingonlypreceding inputdatapoints.SeeFigure1.5(right).Thenumberof timesteps intothepastk‐1,k‐2,…isratherarbitraryforpredictionandsimulation.Thenumberoffuturesteps,however,hasmoreprofoundconsequencesintermsofmodellingrequirements.Figure1.6.Useofnumericalmodels.Left:optimisation.Right:analysis.
numerical model
y(k)
u(k-2)u(k-1)
...
y(k-2)y(k-1)
...
prediction
numerical model y(k)u(k-2)
u(k-1)
...
simulation
strategy numerical model
u y evaluation
optimisation
process y
numerical model
u
y
u
analysis
13
Veryoften,asisthecaseinthepresentstudy,asimulationmodelisusedforoptimisationoranalysis.Whenamodelisusedinoptimisation(Figure1.6ontheleft),thegoalistofindasetof (strategy) parameters for which a certain desired system performance is reached(formulatedasaminimisationormaximisationofapredefinedobjective function).This isdonebyevaluating themodelledoutput signaland looping theoutcomeback toadapt thestrategy, until the desired performance is found. This is also called reverse modelling,becausethereismoreinformationavailableaboutthedesiredmodeloutputthanaboutthemodelparametersontheinputside.Arathersubtleuseofnumericalmodels isanalysis(Figure1.6ontheright).Here,processandmodelcoexistinparallelandthegoalofrunningthemodelistoscrutinisetheworkingof theprocess.Whatcounts isuncoveringdetailsof theprocess, thatarehard tomeasuredirectly, by looking at the results of the numerical model. This is usually more than justcomparingoutputs and .
Figure1.7.Useofnumericalmodelinacontrolscheme.A numericalmodel can be used inside a control loop to facilitate the design of a suitablecontrol measure. Figure 1.7 shows an example lay‐out of a controlled process. It is theperformance of the controller that is the main concern, the extent to which the modelreplicates the process is of secondary importance. There exist different variants of thescheme in Figure 1.7. A last type ofmodel use is fault detection (not shown in a figure),whereseveralversionsofthemodelruninparallelwiththeprocessinordertodetectfailuremodesinasystem.It remains undiscussed so far which model usages benefit from which type of modelconstruction. This all really depends on the problem type, the problem complexity, thespecific questions thatneed tobe answered, aswell as the availabledata and the stateoftheoretical knowledge. Theways computationalmodels canbeused, as introduced in thissection,isnotsynonymouswiththeapplicationofamodel.Howwespeakaboutapplicationsagainquite stronglydependson the focal lengthof thediscipline.A computer scientist orappliedmathematicianmay consider amovingmesh inCFDas an applicationof a certainalgorithm,whileanoffshoreengineerconsiderscomputationsofavibratingmarineriserasan application of a CFD model containing a moving mesh. Section 3.1 digs deeper intoapplicationaspectsofnumericalmodellinginhydraulicengineering.
process
numerical model
u ycontroller
design
r
control
14
1.3 Thesisaims
StudiesintoFIVofgatesuptonowcanbecategorisedinfourgroups:‐ Physical scalemodelling of gate sections complemented by analytical studywith thegoal of gaining fundamental knowledge on excitation mechanisms and optimal gateshapes.
‐ Elastically‐scaled physical scale modelling for checking and improving the design ofactualstructures.
‐Fieldmeasurementsonprototypestructures:monitoringofgatebehaviourforacertaintime period after an incident has happened orwhen there is uncertainty about gateoperationprotocols.
‐CFDsimulationsasapurelyacademicstudyorcomplementarytophysicalmodelling.All four types have proven their value, but are time‐consuming. Given the ongoingdevelopments in computationalmethods and the variety of uses of numericalmodels, anexciting idea is toexaminenewwaystostudygatevibrations.This leadstothreeresearchaimsforthisthesis:Aim1This study sets out to critically evaluate previous research work in the field of gatevibrations, examine new design improvements and measures for prevention andattenuation.Aim2The second aim is to contribute to the development, testing and application of novelcomputational techniques with the purpose of providing the engineer with new tools foraddressing gate vibration issues in design, operation and maintenance. By explicitlydiscussing and interpreting model building and usage, this thesis aims to strengthen thebridgebetweennumericalmodellingandproblemsolving.Aim3Thetopic isablendofresearchareas; themultidisciplinaritycomesontheonehandfromthe combination of hydrodynamics (the flow), structural mechanics and mechanicalvibrations(thestructure),on theotherhand itcomes fromthenumeroussub‐branchesofcomputationalscienceandthelargenumberofavailabletechniques.Thequestionriseshowtodealwithmodelchoiceanduseattheintersectionofdisciplines.The third aim, related to the second, is therefore to explore computationalways to tacklemulti‐disciplinaryproblems that are unsolvable by available conventionalmeans from thedisciplines separately. Looking at it from a reverse angle, the question becomes how toincreasetherelevanceandfind‘killerapplications’foremergingcomputationaltechniques.Beforecommencingtworemarksareinorder.‐Itisnotanaimofthisthesistoassembleandincorporateallpreviousworkandprovideananthologyofthetopic,sincethishasbeendonealreadytosatisfactionintextbooks.Asubstantialamountofaccumulatedknowledge is found inKolkman(1976).Thesameauthor contributed to other, updated overviews, of which Kolkman and Jongeling(1996)isthemostcomplete.ThetwobooksbyNaudascher(1991)andNaudascherand
15
Rockwell(1994)togethercontainawealthofanalyses,usableguidelinesandreferencestofurtherstudies.
‐Theworkdoneforandpresentedinthisthesisisnotrelatedtoanyspecificbarrierinreallife.
Figure1.8.OneofthetwolargesectorgatesoftheSaintPetersburgdam.
1.4 Thesisset‐up
Thechaptersareorganisedaccordingtodifferentusesofmodels.Thefocusfrequentlyshiftsbetweenhydraulicengineeringandappliedcomputerscience.Arun‐throughofthechapters.Chapter 2 introduces the necessary physics background. This makes the computationalchallengemoreclear.Assuchitservesasafoundationfortheapproachesandchoicesmadeinsubsequentchapters.Chapter3goesintonumericalmodelsbasedonphysicsequations.InChapter4,acascadeofphysics‐basedmodelsispresentedforcomputingtheimpactofflowthroughhydraulicgates.ThenChapter5discussesaphysical laboratoryexperimentoftwotypesofunderflowgates.Chapter6proceedswithfundamentalnumericalmodellingofthesame two gates in a finite element model. After this, Chapter 7 introduces a data‐drivensystem for barrier operation that avoids gate vibrations. Chapters 8 and 9 explore recentcomputational techniques in the field of evolutionary computing andmake an attempt toapplythesetothevibrationproblem.Thefinalchapterstatestheconclusionsofthethesis.Table1.1classifieseachchapterforthereader’sconvenience.Shortintermediateevaluationsareaddedwhereapplicable,tokeepaneyeonhowtheresearchfitsintothebiggerpicture.
16
Table1.1.Overviewofchapters. chaptersthesisbackbone 1 10framework/methodology/discussion 2 3 89results fromnumericalexperiments 4 6 7 89resultsfromphysicalexperiments 5
Theintroducednotionsregardingmodeltype(numericalorphysical),buildingofnumericalmodels(data‐drivenorfundamental)andmodeluseenableanotherclassification,asshowninTable1.2.Table1.2.Chaptersaccordingtomodelbuildinganduse.
modeltypemodelbuilding↓
modeluse→ an
alysis
optimisation
control
numericalmodeldata‐driven 9 8,9 7
fundamental 3,4,6 4
physicalmodel 5
17
2 Background
2.1 Introduction
A prerequisite for any type of modelling is understanding the problem, which in turnrequiresknowledgeofthesystem:domainknowledge.Forthatreasonthischapterprovidesabackgroundonthephysicalprocess.Becausetheflowforcesongatesareintimatelylinkedto the flow around the structure, the force analysis first of all builds on a study of localhydrodynamics (Section 2.2). Then, classical linear vibration theory is discussed (Section2.3)andlastlyflow‐inducedvibrationsareintroducedfromtheviewpointofhydraulicgates(Section 2.4). The final section (2.5) draws conclusions from this background study ofconcerntosubsequentchapters.The background is inevitably concise and incomplete due to the shear scope of FIV –referencesaregiventotextbooks.Thetwoothermajorsourcesofdynamicalloadsongatesofhydraulic structures,wavesand ice, are leftoutof this study completely.These requiredisparatephysicalanalyses.
2.2 Hydrodynamics:flowaroundastaticgate
2.2.1 GeneraldescriptionAdischargeregulatingstructureordinarilyconsistsofoneormoresections(alongthecross‐flowdirection)withuniformcrosssection.Forgatesofsufficientspan,theinfluenceofthesides (piers) aswell as the presence of an adjacent gate can be neglected. Therefore it iscustomtoschematiseavertical‐liftgateforunderflowdischargebytakingitscrosssectioninthemainflowdirection.Theareaofstudyishencethetwo‐dimensionaluprightplanealongthestreamwiseaxisoftheflow.ThisgivestheflowmodelsketchedinFigure2.1,acanonicalsituationinfree‐surfacehydrodynamicsfoundabundantlyintextbooks(e.g.Battjes2001).
Figure2.1.Definingschematisationofsubmergedflowpastavertical‐liftunderflowgate.Theflowisfromlefttoright.
h3
connection to barrier structure
Сс a
h2
h1
z = 0
gate body
a
H2H1
αiUi2g
2
H1
18
ThemainflowfeaturesofthesteadyflowsituationwithafixedgatepositionareshowninFigure2.1.Thisfigureillustratesthesubmergeddischargestatethatischaracterisedbysub‐criticalflowdownstreamofthegate.Thedesignation“steady”referstothestationarystatewhere the flow pattern, consisting of the pressure scalar and flow velocity vector on allpositionsinthefluid,doesnotchangewithtime.Thisfigurewillreturninalternativeforms,thusplacingemphasisondifferentaspectsandschematisations.There are twodrops in energy level∆Hi,which areoften lumped intoonehydraulic headdifference∆h=h1–h3.Thereisazoneofrecirculationdownstreamofthegate,characterisedbyalargeeddywithahorizontalaxisperpendiculartothemainflowdirectionandreverseflownearthesurface.ThespecificsituationinFigure2.1showssubmergedflow,wheretherecirculationexistsrightnexttothegate,incontrasttothefreeflowconditionwherethereisa detached and more violent ‘recirculation’ away from the gate, a hydraulic jump. Therecirculation zone under submerged discharge is also called surface roller, as thefluctuationsaffectthefreesurface.Thelocalloweringofthefreesurfacebehindthegateisduetotheacceleratedflow.Asthevelocityprofilegraduallyreturnstoafullypositiveprofilewith parallel streamlines (this is the case at h3), the free surface also recovers. Thedownstreamregionissometimesreferredtoasa‘wake’,althoughthistermistraditionallyreservedfortheareadownstreamofbodiesthathaveflowonbothsides.Thereisacrosssectionintheseparatedflowlayer(oremergingjet)atsomedistanceofthegatedownstreamwherethe thicknessof this layerabovethe flooror flowbed isminimal,this height is the vena contracta. At this point, we define hvc = Cc*a, where Cc is thecontractioncoefficient.Beyondthispoint, theflowdivergesanddecelerates.ThedischargeformulaofflowpastthegatethatintroducesthedischargecoefficientCdiswritten
2 . (2.1)
Continuity or mass conservation has the general form qi = constant = q. For the sectionbetweenh1andthevenacontractathisbringsthecontractioncoefficientCcintoplay:
1 1 , with1
, (2.2)
whereUi is thedepth‐averagedstreamwisevelocity incrosssection i.Allbasic insightsonfastchangingflowrelyontheprincipleofBernoulli,whichstatestheconservationofenergyas
constant,with . (2.3)
Here,Histhetotalheadorenergyhead,histhepiezometricheadorhydraulichead,zistheelevation above an arbitrarily chosen datum (most conveniently the bed), p/ρg is thepressureheadorstaticpressureandu2/2gisthevelocityheadordynamicpressureterm.Alltermsweredividedbyγ=ρg,thespecificweightofwater,sothatthelengthunitisacquired.Thesubscriptihereindicatesanarbitrarylocationinthefluid.equation2.3inotherwords
19
saysthatthetotalenergyinapointinthefluidconsistsofenergyfromtheheightofthefluidrelative to a datum z = 0, plus energy from pressure, plus kinetic energy from the fluid’smotion.In the following,we adopt the approach byNaudascher (1991). Application of Bernoulli’sprincipletothesectionofflowaccelerationbetweenh1andhvc,gives
2 2
∆ , (2.4)
whereαi are coefficients for thenon‐uniformityof thevelocitydistribution,Ui isagain themeanvelocityoverthedepthand∆H1istheenergylossorheadlossfrom1to2relatedtotheformingofboundarylayersalongthebedandtheupstreamgateplate(duetodownwardflow)andrelatedtothesmallcornereddyattheupstreamsurface.Inthedecelerationpartofthesubmergedflowinsection2‐3,anunknownandmuchmoresignificant amount of energy ∆H2 is lost (∆H2 >> ∆H1) and therefore applying Bernouli istroublesome. The momentum equation is used instead, which gives, after assuming ahydrostaticpressuredistributioninthevenacontracta:
2 2
, (2.5)
withqthedischargeperunitwidthin(m3/s)/morm2/s,athegateopeningorgateheightandβ, likeα,auniformitymeasureofthevelocityprofile.ForhighReynoldsnumberflows,theviscouseffectsthatgiveriseto∆H1>0andα,β<1arenegligible(Naudascher1991),sothat it is safe to assume ideal flow conditions for real‐life situations rather than forlaboratorysettings,using∆H1=0andαi=βi=0.The free flowversionof theabovegoesequivalentlyand issimpler,since thedownstreamwaterdepthafterthedecelerationh3doesnotinfluencetheconditionsnearthegate–andhencealsonotthedischarge.ThefollowingrelationbetweenCdandCccanbederived:
1 ⁄
; (2.6)
CcisanempiricalparameterfromwhichCdcanbededuced.Thisapproachismorerespectfultowardsthephysicalcausesfordifferentdischargesthansimplifiedtreatmentsthatmentionone discharge formula for submerged flow ( 2 ), and one for free flow( 2 ),where is thesingleempiricalcoefficent,usually ≈0.6 is taken for freeflow. In more elaborate discussions, variations of the parameter Cc are investigated andexplainedfromphysicalprinciples.Mostgenerally,Ccisafunctionofa/h1,thegategeometryandtheflowbottomgeometry.Inthelimitingcaseofagatethatisalmostclosed, → 0,thecontractionisthesameforfreeandsubmergedflowandCc=Cd.For0<a/h1<1,Cc=f(a/h1)hasapositiveandincreasingslopeforsubmergedflow,whileCcvariesmuchlesswitha/h1forfreeflow(seediagrambyRouseinNaudascher,1991).
20
For free‐surface flow themomentumbalance can be set up for the complete stretch overwhich the deceleration takes place, that is, on either side of the hydraulic jump. Thedownstreamwater levelpast thehydraulic jumpcanbederived fromthis. Inaddition, thelossofenergy∆HinthehydraulicjumpcanbefoundusingtheBernoulliequation.Onlyforspecial situations, depending on slope type, bed roughness and boundary conditions,analyticalcomputationscanbemadeof the locationof the localwatersurfaceatarbitrarydistances (not far) from the gate (see e.g. Chow, 1959 and Van Rijn, 2011). Usually suchcomputations, the Bélanger equation is an example, already involve iterative schemes, sothattheycanhardlybecalledanalytical.2.2.2 TurbulenceAmore realistic description is different from the fully steady case first of all because thewake circulation region always contains some time‐dependent aspects: most notablyvorticesshedfromthegateanddisturbancesindownstreamwaterlevel.Therefore,theareadownstream from the gate is sometimes said to be quasi‐steady, alsowhen the upstreamconditionsaresteadyandirrespectiveofthedownstreamflowregime.Unsteady flowaspects are an inherentproperty of gate flowbecause of the occurrence ofturbulence.Twodifferentsourcesofturbulenceenterthepicture:freeturbulencefromtheshearinthemainflow(causedbythepresenceofthegateactingasanobstruction),andwallturbulencefromshearduetofrictionalongtheflowbedandgatebottom.Thisresultsinthecombinedpresence of a boundary layer adjacent to thebottomboundaryof the flow (thefloor) and a free shear layer under and/or directly past the gate. The extent to which aboundary layer develops along the bottom of the gate entirely depends on the surfaceroughness,thelay‐outoftheprofileandthemainflowvelocityofthedischarge.Anyway,theturbulenceoriginatingfromtheboundarylayerandfreeshearlayergovernsthetransitionalzones above and under the accelerated main flow under the gate and/or directlydownstreamfromit.Gateprofileswithasharpedgeontheupstreamside,suchastherectangulargateinFigure2.1,causeseparationoftheflowatthatedge,the‘leadingedge’.Flowseparationmeansthattheflownolongerneatlyfollowsthecontoursofawall,buthasenoughmomentumtoleavethe boundary. The pressure on this boundary is lower in the upstream part before theseparationthaninthedownstreamsectionaftertheseparation.Thisdownstreampartoftencontains irregularities (reversed flow, possibly part of an eddy) that are not found in themain flow. Figure 2.2 gives a few examples of gate bottom profiles with the edge of theseparation layer indicated. Round gate bottom profiles portray less predictable flowseparation. Again depending on the precise geometry,material roughness andmean flowvelocity, the flow may separate at different points along the curved gate surface. Moreformal: the separation point is in this case dependent on the Reynolds number. Thesefeatureshavebeenstudiedbyvariousauthors(e.g.Hardwick,1974;Vrijer,1979)toimprovegatedesigns.
21
Figure2.2.Examplesofgateprofileswithflowseparationindicated.AfterdesignstudiesbyVrijer(1979).At certain approach flow velocities and given the right geometry, the flow separation canexhibit strongly periodic detachment of rotational flow structures: vortex shedding. Allconstructiondetailsinthegateedgeregionarefactorsintheproductionofturbulence.Forinstance, theplacementof a rigidhorizontal girderon lowerdownstreampartof the gate(meant to reduce bending of the gate) could trigger entrainment of flow from therecirculationzone into theshear layerof the submerged jet;Naudascher (1991)mentionsthis as a cause for a permanent low‐pressure zone (under this girder) that influences thestaticliftforce.Calculations involving turbulence start with separating the instantaneous flow velocity inonepointintoatime‐averagedcomponentandafluctuatingcomponent.ThisistheReynoldsdecomposition, ,with 1 ∗⁄ ∙ d overa time length t*.Naturally, the samedecomposition is done for the other dimensions of the velocity vector. The turbulenceintensityorturbulencelevelImeasures,quiteliterally,theintensityofthefluctuationsofthevelocityinonepointasapercentageofthemeanflow.Bydefinition,
′ , (2.7)
andviceversa fortheothervelocitydirections.Theoverbardenotestime‐averagingagain.The turbulent kinetic energy (TKE) is a scalar that represents turbulence strength of theentirevelocityvector,itisdefinedas 1/2 u′ v′ w′ .Omnipresentflowinstabilitiesmaketurbulenceatrulythree‐dimensionalphenomenonwithdifficult‐to‐predict time‐varying patterns (Tritton, 1988). There is a risk of ‘analysisparalysis’(Mahajan,2010)whentryingtostudyallphenomenafortheplethoraofpossiblelay‐outsandconstructiondetails. Forexample,howdo the features changewhen thegatemoves to a new position? And does it matter how fast this movement goes? SomemorediscussiononthisfollowsinSection2.5.2.2.3 HydrodynamicforcesonastaticunderflowgateFigure2.3showsthestaticforcesonarectangularvertical‐liftgate.Ontheleft,theforcesareshownschematically,frictionisleftout.Ontheright,astaticsdiagramforthehorizontalloadisgiven.
22
Figure 2.3. Left: Free body diagram of gate body for static force equilibrium.Right: gateas verticallyplacedbendingbeamwithhorizontalpressures(h1andh2asdefinedinFigure2.1).First the easier‐to‐estimate horizontal force on the gate are discussed by considering thetotal force on the upstream gate plate perpendicular to flow direction. The hydrostaticpressuredistributionp(z)=ρg(h‐z)givesaneasilycomputedupperboundforthehorizontalflowforce.Itcanbeusedforfreeflow(usingtheupstreamwaterlevel)andforsubmergedflow (subtracting the pressure profile of the downstream water level). This leads to aconservativeestimatebecausethevelocityheadoftheacceleratedflowupstreamhastobesubstractedfromtheintegratedpressure.SeeFigure2.3(right).The force exerted by the flowing water on the gate is found more accurately from themomentumbalance between the upstream section and the vena contracta. Intuitively, thebalancestatesthatthetotalflowforceupstreamequalstheforceonthegateplusthetotalflow force at the downstream point of maximal vertical contraction. Mathematically, thislookslike
,12
1 1(2.8)
forsubmergedflow.Thefreeflowversionofthisequationfollowsfromreplacingh2withCca.After thehorizontalhydrodynamic forceshavebeenestablished,horizontal support forcesare subsequently determined from the applicable structuralmechanic (statics) principles.TheschemeinFigure2.3assumesastaticallydeterminedsupportcombination,sothattwosupportforcescanbecomputedinadirectmannerfromtheforceandmomentequilibria.The vertical hydrodynamic force, or lift force, requires more analysis. Following againNaudascher(1991)becauseofhisrigorousandgenericapproach,thelocalpiezometric‐headcoefficientChanditsspatialaverage alongtheundersideofthegateareintroduced.ThedefinitionsarederivedfromconvertingtheBernoulliequation,appliedtoasectionbetween
support element
gate body
Flift
Fg
Fsup,vert
FArch
Fw,horiz
flow
Fsup,horiz
aρg(h1-a)
ρg(h2-a)
p(z)hydrostatic
hydrodynamic
U 2g2
23
a point i at the gate bottom and the head in a point of reference downstream, intodimensionlessform(afterdivisionbythevelocityhead):
/2
and 1
, (2.9)
whereD is the thickness (sometimes rather confusingly called depth) of the gate body instreamwisedirection.Thesedefinitionsaresuitable forsubmergedflow,h2 isusedhereasreferencehead.The lift coefficient canbepositiveornegative,dependingongate type,gate opening and water levels, and thus contribute to downpull (suction) or upward lift.Figure2.4showsthedefinitionsforagatewithacurvedundersideandclosedprofileandisusedtoexplainthebasiccomputationoftheliftforce.
Figure2.4.Computingthestaticliftforceforsubmergeddischarge,afterNaudascher(1991).Thefigureattemptstoillustratethatthetotalverticalforcearisingfromthegatebeingpartlysubmergedinflowingwaterconsistsoftwoparts:thetrueliftresultingfromthe(non‐zero)flowpast thegatebodyand theArchimedeanorbuoyancy forceofdisplacingavolumeofwater. For submerged flow, for the example profile in Figure 2.4, both components areupwardandthetotalforceis
2
, (2.10)
whereB isthewidthofthegateinthehorizontaldirectionperpendiculartothemainflow(recallingthatauniformgatesectionwasassumed).Thesecondtermis,forsubmergedflow,equaltotheweightofthedisplacedwaterunderthereferencehead.Now,usinginsteadthegateopeningaforreferenceinthecaseoffreeflow,itisfoundthatthisArchimedeantermis
Сс a
h2
h1
a
x
zi
pi
D
ρg
Ui2g
2
Uvc
Uvc2g
2
24
negative, as a – zi < 0 for all i along the underside. Thus it becomes a downward forcecontribution.The coefficient can be estimated from potential flow analysis only for non‐separatingflow,whichisahugelimitation.Soingeneralthecoefficient,thatdependsona/h2,thegatetypeandtheflowbottomgeometry, isderivedfrommeasurementsor fromdedicatedCFDsimulations.Theseparationintohorizontalandverticalforcesonagateisgenerallynotastrivialasfortherectangular,purelyverticallyplacedgate.Forthecurvedgateplateofataintergate,forinstance, an extra essential point is to consider themoment of the resulting force (forcerelative to pivot). Moments that tend to close the gate are known to cause self‐excitingvibrationswhenthereisinsufficientmass‐damping(Naudascher,1991).Avastamountofempiricalknowledgeonliftforcesfordifferentgatetypescanbefoundintextbooks for many sorts of hydraulic gates and local bottom profiles (sills, aprons etc.),givingreasonableestimatesof thestaticgate forces.Whenthe impactofvaryinghydraulicconditionsisconsidered,thingsbecomemoredifficult.Thedynamicforcesduetovibrationscanbeseenasanextremecaseofvariableconditions.2.2.4 Navier‐StokesequationsMany books about hydraulic engineering have beenwrittenwithout explicit treatment orevenmentionoftheNavier‐Stokesequations.WheneverCFDis involved,however,thereisnowayaroundthem.Heretheyare,inconciseform:
∙1
. (2.11)
Hereuisthevelocityvectorinm/s,ρiswaterdensityinkg/m3,pispressureinPa,νisthekinematic viscosity in m2/s, is the derivative operator and denotes the advectionoperator. Note that for free‐surface flows of course a gravity term should be added asexternalforceintheverticaldimensionontherighthandsideof(11).Thefluidofinterestiswater,which isassumedtobe fully incompressible,so that thedensitydoesnotvarywithpressure.Themassconservationorcontinuityconditionisthus
∙ 0. (2.12)
In chapter4and6 theseequationswillbeevaluatednumerically, after substitutionof theReynolds decomposition defined in Section 2.2.2 for all velocity terms. For now, it isinteresting to realise that the terms responsible for turbulence, theReynolds stresses, aremorethanonehundredtimeslargerthantheviscousstressesundernaturalflowconditionsofRe>104,say(VanRijn,2011).
25
2.3 Linearvibrations
2.3.1 BasictheoryTurning now to the structural schematization, the same cross section is considered fordescribingthestructuralresponse.Thegatebodyisthoughttoactasanisolated,rigidpointmassconnectedtoafixedstructure.Thesupportingconnectionisthoughttohaveacertainstiffness and to provide damping. Thus the structural response is modelled as a familiarmass‐damper‐springsystemwithonedegreeoffreedom(1d.o.f.orsingledegreeoffreedom,SDOF),seeFigure2.5below.Whatmakesthesituationspecialisthatthebodyresonatorispartlysubmergedinafluidandthatthefluidis(bydefault)inmotion.
Figure2.5.Adampedmass‐springschematizationofahydraulicgate.Thisschematisation isconsistentwithFigure2.1,onlythistimetheemphasisisonthestructuralresponse.
Thissectiongivesaselectivebackgroundonmechanicalvibrations.Theclassicandpracticaltextbooks by Den Hartog (1956) and Piersol and Paez (2010) are used throughout thissection.Other references are givenwhere applicable. The situationof a dampedvibrationwithsinusoidalforcingistakenasabasecase.Itisassumedthatthespringislinear‐elastic,the damping is viscous and that themass is concentrated in the gate (i.e. the spring anddamper have no mass). In addition it is assumend the gate is in static equilibrium. Thismeansthatthegate’sownweightandthestationaryliftforceexertedbytheflowingwaterarebalancedpreciselybyastaticsupportforceFsup,stthatisleftoutofFigure2.5forclarity.That is, the static deflection ⁄ is not implemented in the analyticaldescription;thestaticequilibirumpositionsimplycorrespondstoy=0.Induecourseitwillbecomeclearthatalltheseassumptionsareinfactdisputable.InFigure2.5twofreebodydiagrams(FBDs)arediscerned.FBDIdescribestheverticalforceequilibrium of the gate body itself, which follows the second‐order ordinary differentialequation(ODE)ofmotion:
. (2.13)
rigid gate body
flow
y(t)
Fw(t)
m
ck
FBDII
FBDI
26
Here,yistheverticaldisplacementrelativetothestaticequilibriumposition,misthemass,cis the damping coefficient, k is the stiffness coefficient, Fw is the time‐varying waterexcitation force equal to the boundary integral of the instantaneous pressures at the gatebottom profile. Newtonian notation is used: the overhead dots indicate derivations withrespect to time. The other force equilibrium FBDII in Figure 2.5 concerns the dynamicequilibriumofthesupport
, . (2.14)
IfweassumeforthemomentthatFw(t)=0,thevibrationiscalledfreeorunforcedandthedescription of FBDI is as follows. Introducing the natural undamped frequency
⁄ 2 0,thecriticaldampingcoefficient 2 andthedampingratio ⁄ ,thesolutiontoequation2.13for0< <1(less‐than‐criticaldampingorunderdamped)canbewritten
⁄ sin sin , (2.15)
where is the damped natural frequency given by 1 and C and θ areconstants. The logarithmic decrement is defined as ln ⁄ , where yn is thedisplacementamplitudeofthen‐thpeak.Bysubstitutionforn=1itisfoundthat
2
1. (2.16)
This implies that for small damping ≪ 1, we have 2 . Returning to the forcedvibration, if theexternal forcing isa sine sin , themotionequation2.13 canberewrittenas
2sin
. (2.17)
The amplitude of the mass motion is called the motion response and follows from theparticular solution of this ODE. The general solution of (2.17) corresponds to the freevibrationwithsolution(2.15),butintheunderdampedcasethenaturalfrequencyvibrationdiesoutandonlythesteady‐stateforcedvibrationwithangularfrequencyωremains.Thissolutionis
⁄
1 Ω 2 Ωsin ,
with tan and ⁄ . (2.18)
27
This resultmakes clear that the phase angleθ between the force and thedisplacement isnon‐zeroonlyifthereisnon‐zerodamping.Additionally, itcanbeshownfrom(18)thatinthelimitingcases ≪ and ≫ ,y(t)tendstooscillate inphasewithFw(t), i.e.θ=0,whereas implies that θ = ‐π/2which means that the motion is out of phase: itprecedestheforceby90degrees.Therefore,analternativewayofwritingequation2.18canbeadoptedthatdecomposestheformulainonesineandonecosinecomponent:
⁄
sin cos . (2.19)
WhereRisthedimensionlessresponsefactorandRinandRoutarefunctionsofΩand .InallaboveexpressionsthesimplerequationsofundampedSDOFvibrationsfollowtriviallyasaspecialcasefromc= =0.LookingnowatthesupportforceFsup,dyn(FBDIIinFigure2.5),themagnitudeofthisvectorisfoundbyrealisingthatitstwocomponents,thedampingandstiffnessforceinequation2.14,areoutofphaseby90degrees,whichyields
, . (2.20)
ItiscommontodefineaforcetransmissibilityquantityTas , ⁄ ,fromwhichitispossibletodeducethat
,
sin ,
with and2
1 2. (2.21)
Inotherwords,theforcetransmissibilityTandaccompaniedphaseΨarebothfunctionsofthefrequencyratioΩanddampingratio .
28
Figure 2.6. One of the 34 gates of theHaringvliet barrier in TheNetherlands. This is a tainter valverotatingaroundahorizontalaxis;ahingedsupporttruss(thetriangularelementvisibleinthetopofthepicture)providesthemovement,itisdrivenbyalargehydrauliccylinder.PotentialvibrationsinthesamerotationdirectioncanbedescribedbyaSDOFsystem.PicturebyDeltares.2.3.2 FrequencydomainInvibration theory, sooneror later it isnecessary toswitchtocomplexnotation. Itmakesgeneralisations and extensions of the analysis more compact while maintaining physicalinsights. Fully in line with the definitions introduced earlier we may rewrite
cos sin and assume a solution to 13 of the formcos sin fortheresultingdisplacement.Therealpartsoftheseexpressionsgive
theactualexcitationandresponse,respectively.The imaginarypartofy(t)givesthephasedifferencebetweenforceandmotion.Theequivalentofequation2.18becomes
1⁄
1 2, (2.22)
where now in the last step the mechanical admittance X is defined, a function with thefrequency as independent variable. Notation from Naudascher and Rockwell (1994) islargely followed. From the insight that the absolute value of X(ω) is the ratio of theamplitudesy0/F0,whichisthequotienttermhiddeninequation2.18,itfollowsthat
| | , (2.23)
with θaccording to equation2.18 and |X(ω)| is called the complex frequency responseorreceptance.Substitutioninequation2.22thengivesthedesiredsolution
29
| | . (2.24)
AnobvioustrickistouseFourieranalysistomaketheextensiontoarbitraryperiodicforcingfunctions.ThefirststepistodecomposeFw(t)bywritingitasaFourierseries.Aslongasthemotionequationislinear(thatisweassumealineartime‐invariantorLTIsystem),thetotalresponse isequal tothesumoftheresponsesof individualharmonics(multiplesofabasefrequency). This always gives a periodic response y(t). A further extension is to considernon‐periodicexcitationfunctions.HeretheFourierintegralcanbeusedfordecompositionofthe force signal as a function of the frequency. Switching now to the frequency f=ω/2π,more common for spectra, the spectraldensitiesS of the excitationFw and the responseyobeytheequation
∙ | | (2.25)
The power spectral densities have the useful property that the integral over the entirespectrumequalsthesquaredvalueoftheroot‐mean‐squarevalueofthesignal.
2.4 Physicsofflow‐inducedgatevibrations
2.4.1 Introductiontoflow‐inducedvibrationsFluid‐structureinteraction(FSI)isawideandactivefieldofresearchwithahugenumberofapplications. It can be interpreted as an umbrella term for physical systems where themutualdynamicsof fluidsandsolidsplaysacentral role. Incomputationalcontexts,FSI isoftenreservedforproblemswherethestructurehasacertainflexibilitytodeform,suchthatthechangesofthefluid‐solidinterfaceintimeandspacearethemaininterest.Itisimportantto be aware of the fact that different properties of fluids (viscosity, density) and solids(stress‐strain, d.o.f.s)make for drastically different FSI problems. A numericalmodel of aflapping airplane wing may be fully calibrated and have high accuracy; it will in manyrespectsbeuseless as abasis for amodel of apontoon floating inwater. Examplesof FSIproblems that illustrate these ranges are the bending of slender structures in flow (e.g.airplane wings), bloodstreams inside the body, and wind‐induced motion of high‐risebuildings.Païdoussisetal.(2011)giveageneraltreatmentofFSI.The field of flow‐induced vibrations (FIV) focuses on the motion of objects in flow withlimited degrees of freedom. The solid is an object that is too stiff to have the flow causewave‐like phenomena on the fluid‐solid interface. Rather, the object undergoes atranslational,rotational,orbendingmotionoracombinationofthesethree.Thisiswhythepresentedmass‐springmodelisasalientfundamentforFIVproblems.Authorativetextbookson FIV are Blevins (1990) andNaudascher andRockwell (1994). The classical case studyobjectinFIVisacylinder,definedasasolidobjectwithoneofitsthreedimensionsuniformandelongated,andwhichcanhaveallsortsofcross‐sectionalshapes,themostfamiliarbeinga circle anda square.All objectswith considerableblockingof the flow–characterisedbyflow separation for flow from all angles– are named bluff bodies, in contrast to aerofoilswhichhaveastreamlinedwing‐typeshape.
30
Experiencesfrompracticewithreal‐lifeprototypedataandevaluationofFIVcausesformaninvaluablecategoryofFIVstudies.Aspartofmanyinvestigationsintodynamicsofhydraulicstructures in The Netherlands built in the second half of the 20th century, Kolkman andJongeling(1996)giveasizeableaccountofFIVwithampleattentionforgates.Morerecently,Kanekoetal.(2008)compiledengineeringexperiencesofFIVfromJapan.Becausetherewasand is often a strong consciousness only for preventing vibrations from growing beyondsafetylimits,in‐depthphysicalanalysesanddevelopmentofgeneralcomputationalmethodshavenotreceivedtoomuchattention.Vortex‐inducedvibrations(VIV)dealwiththoseflow‐inducedvibrationsof(cylindrical)bluffbodieswith flow on both sides. The canonical case in hydrodynamics of the Von Karmanvortex street of flow around a circular cylinder can be extended by giving the cylinderfreedom to move in cross‐flow direction; as such it has become canonical in VIV too.Academic research in this area has been driven enormously by industrial applications innuclear power plants (cylindrical heat exchangers) and in offshore engineering (e.g.pipelines,marinerisers).Moreover,thetwocanonicalcasesmentionedarefrequentlyusedas benchmarks for computationalmodels owing to the complexity of the physics and theavailability of measurement data. A frequently cited overview of analytical and to someextent also numerical models of VIV is given by Sarpkaya (2004). Gate vibrations are adistinctsubsetofFIVwithlittleoverlapwiththeVIVsubset.Gatesareessentiallydifferentbecausetheflowexistsonlyononesideoftheobjectandthereisasignificantinfluencebythepresenceofatleastonewall.The remainder of Section 2.4 uses and combines knowledge found in Blevins (1990),NaudascherandRockwell(1990)andKolkmanandJongeling(1996).2.4.2 DimensionlessparametersAgoodwaytostartisbyidentifyingthemostrelevantdimensionlessparameters.Table2.1liststhesealphabetically.Table2.1.Dimensionlessparametersofgatevibrations.
symbol definition
dimensionlessamplitude ‐ ⁄
dampingratio 2√⁄
Froudenumber Fr ⁄
massratio mr ⁄
reducedflowvelocity Vr ⁄
Reynold’snumber Re ⁄
Scrutonnumber Sc 4
Strouhalnumber St ⁄
turbulenceintensity I ⁄
31
Inthetable,gatethicknessD isusedascharacteristic lengthscale,exceptforFrwherethewater depthh is used. The standard deviation ofu, denotedσu, is used in the turbulenceintensity formula. The difference betweenVr and St is important. Both are dimensionlessvelocities or, alternatively, dimensionless frequencies. The reduced flow velocity Vr dealswith response: it uses the response frequency and 2 ∆ as characteristic velocity,whereastheStrouhalnumberdealswithexcitation:itusestheexcitationfrequencyandthesteadyundisturbedupstreamvelocity as characteristic velocityU. TheStrouhalnumber islinked with periodic excitation or forcing from vortex shedding. Its numerator can beinterpretedasthespeedatwhichdisturbancesfromseparationattheupstreamedgemovealong the bottom gate boundary. The reduced flow velocity indicates the velocity of flowunder the gate relative to the gatemotion, similarly it is a dimensionlessmeasure of thefrequencyofthegatemotion.Themass ratioand thedampingratioaredecisive for theactual response that is realised.Themass ratio gives themass per width unit relative to the surrounding fluid and is aninertial measure of how easily the gate body initiates into motion. The Scruton numbercombinesmassanddampingratios.ItispopularlyusedinVIV,buthasthedisadvantagethattheeffectsofmassanddampingarenolongerdistinguishable.Specificallyforgates,dampingisofsomuchimportancethatonewouldliketoquantifyitseparately.TheFroudenumberdescribestheinfluenceofthefreesurface,notinallcasesdirectlyimpactingthevibrations,butneverthelessrelevantforfreeflowandintermediateconditions.The infamous Reynolds number is in this context mainly relevant in physical models forensuring that viscous effects are relatively small. Of course in physics‐based numericalmodelling, thesuccessofanyapproach(other thandirectnumerical simulation,DNS) thatattempts to estimate the Reynolds stresses depends strongly on Re. The higher Re, thebroadertherangeoflengthandtimescalesintheflow.Themost conciseway to summarise all FIV efforts is to state, as an adaptation toBlevins(1990),thatweaimtofindtheamplitudeasafunctionoftheotherparameters:
, , , Fr, Re, St, , (2.26)
where the Scruton number is left out because it is a function of the mass ratio and thedampingratio.Fromareversepointofview,weare justaswell interested indeterminingthefunctionf,givenvaluesofthedimensionlessparameters.In addition to providing general physical insight, the dimensionless parameters have astrong importance in thedesign, interpretationandcomparisonsofphysicalmodels in thelaboratory.Whenscaled‐downmodels(physicalandnumerical)arecomparedtoprototypestructures, the dimensionless parameters are ideally scaled by equal factors –somethingwhich is quite impossible in most circumstances– in order to keep scale effects to aminimum. In particular, for dynamics of structures, scaling of elasticity and dampingrequiresmeticulousefforts.
32
2.4.3 Causesofflow‐inducedgatevibrationsVariouscategorizationsofFIVarepresented in literature.Theemphasis inthemostusefulcategorizations is on the causes of vibrations, which are described in detail in excitationmechanism theories.Acomplicating factorwhenbrowsing through literature is thatmanypast analysesweredonebasedonunjustifiedassumptions regarding excitations.Also, theterminologyregardingvibrationcausesisnotundisputablyagreedupon.Here,theexcitationcategoriesareintroducedbyFigure2.7.
Figure2.7.Causesofflow‐inducedvibrationsofhydraulicstructuregates.The proposed subdivision in Figure 2.7 is close to that byKolkman and Jongeling (1996),withtheonlyadditionthat(i)and(ii)aretreatedseparately.Pointsofdiscussionleadingtoother subdivisions have to do with how turbulence and flow instabilities are defined. Inliterature, mechanisms are sometimes attributed to the classes movement‐inducedexcitation (MIE), instability inducedexcitation (IIE)and impinging leadingedgevibrations(ILEV) introduced by Naudascher (see Naudascher and Rockwell, 1994). Although thesehavetheirvalueasshorthandsforsomethecausesdiscussedhere,andtheywillbeusedinChapter 5 and 6, they are together not all‐inclusive for all encountered phenomena.Excitations (i)–(v) are briefly discussed only up to necessary detail for further work; in‐depthdiscoursesarefoundinthetextbooksalreadymentioned.(i).ExcitationbyturbulenceHydraulicgatesareconstantly incontactwith flowingwater.This factalone isasourceofexcitationbecausetheincomingflowisinrealityneverfreeoffluctuationsaroundthemeanvalue (see equation2.7); I ≈10% is already consideredhigh‐level turbulence). In an idealReynolds decomposition, all periodicity is filtered out, such that u’(t) is a noisy, randomsignalwithauniformfrequencydistribution.Thedynamicforceonthegateassociatedwiththis excitation is said to be caused by external turbulence excitation, assuming that thesourceof the turbulence liesoutsideof thestructure, that is, thenon‐periodic fluctuationswere alreadypresent in the flowbefore reaching the structure. Computationally, equation2.25inSection2.3forrandomsignalexcitationisapplied:
turbulence stable vortex
shedding
flow instabilities
(i) (ii) (iii) (iv)self
excitationunstable
fluid resonance
(v)
33
∙ | |
Figure2.8.Illustrationofaresponsespectrumderivedfromaspectrumexcitationfora(gatestructure)systemwithtwoeigenfrequencies.Figure 2.8 gives a devised example of computations in the frequency domain. When thenatural frequenciesof thesystemX (determinedby thegateand its suspensionstructure)are known, the response spectrum Sy is found through multiplication with the forcespectrumSF,w.TheparticularillustrativeexampleofFigure2.8showsthattheresponsetoabroad‐bandexcitationrevealsthenaturalfrequenciesofthestructureintheoutputsignal.The occurrence of low‐level, irregular loading on a hydraulic gate is an everydayphenomenon, hardly noticeable and not harmful to the structure. This is a passiveinteraction:theresponses(displacements,deformations,stressesandsupport forces)haveno influence on the dynamic loading. Excitation by this random‐type turbulence leads todynamic forces of just a few percent of the stationary lift force, Kolkman and Jongeling(1996)mentionamaximumof10%.(ii).ExcitationbystablevortexsheddingThe mere presence of the gate is an interference of the incoming flow by a bluff body,creating disturbances of a periodic nature in the form of vortices and eddies, see Tritton(1988). If the sizes of these periodic parts of the flow (‘coherent structures’) areunambiguouslyrelated to thesizeof theobject fromwhich theyareshed(acharacteristiclength scale of the gate, commonly its thickness D), and their propagation speed isproportionaltothemeanincomingflowvelocityU,thentheStrouhalnumberisaconstantandhenceuseablefordescribingtheexcitationfrequency.Keytothisconditionistheshapeon the upstream gate side that is decisive for the type of flow separation. As depicted inFigure 2.2, a sharp leading edge fixes the point of separation, resulting in stable vortexshedding,whereasroundelementsmaketheseparationpointdependentonRe(evenathighRewhenviscosityisnofactor),resultinginunstablevortexshedding,treatedunder(iii).Elaboratingonthestable typeofvortexshedding,a thickenoughgatebottomwill feel theperiodicimpactoftheseparatedlayerasitperiodicallybendsbacktothegate’sundersideatsmall gate openings. This is called impingement (ILEV). Theperiodic excitation force is inthiscasefoundfromSt.Thereareflowmeasurementdatagiving
St=f(a/D,geometry)
f
SFw
f
| X |2
f
Sy
34
foranumberofgate types thatgivestable separation.Togrowsome intuition, consideragateatasmallopening,saya/D=0.4,withinitiallyzeroheaddifference∆h.Astheupstreamwater levelslowly increases,∆handUwill increase,and–accordingto theconstantSt–sowill the vortex shedding and excitation frequency fexc (=ω/2π in Section2.3). The criticalquestionisnowwhetherUcangrowsohighthatfexcapproachesf0.Thiswillultimatelyleadto resonance amplitudes dictated by the damping in the system. To stay away from thissituation,thestiffnessoftheconstructionelementsispreferablyhighenoughtohaveatleastf0 > 2fexc,max, where fexc,max corresponds to the maximum flow velocity U. This kind ofanalyticalmodel is called a vortex‐sheddingmodelbyTondl etal. (2000), theyuse it as abaseforvariousvibrationanalyses.GiventheknowledgeonSt‐valuesandabovedesigninstruction,significantvibrationsduetothisexcitationcanbeavoided.Notethattherelationbetweenflowfieldandgategeometryisnotalways‘neat’,therecouldbeabandofexcitationfrequenciesratherthanasinglefexcforagivenapproachvelocityU.Thisisnosurprisebecausethesheddingofvorticesisprincipallya turbulent effect arising from flow past the structure (a kind of ‘internal turbulentexcitation’asitwere,althoughthistermisneverused).TheStrouhalnumbercaninthiscasestillbeappliedtothedominantfrequencyofthisband,however.Inconclusion,turbulence‐relatedforcingbyexcitations(i)and(ii)isunlikelytocausevibration‐relatedtrouble,aslongas the structural eigenfrequencies indeed lie outside the spectral band of these excitationfrequencies.(iii).ExcitationbyflowinstabilityForunderflowgates,threeflowinstabilitycasesarerelevant:‐ The separation point does not have a fixed position if it depends on the inflowconditions(Re),thisisthecaseforaroundedleadingcorner(Figure2.2).Thisgivesrisetoan instabilityof theseparatedflowasawhole.Thepressuredifferenceonthegateinherenttotheflowseparationisnowtime‐dependent.Thisisitselfalreadyaperiodicforcing.
‐ The separated flow after some distance slowly starts losing its momentum due toexchangeswith the surrounding flow (by formation of tiny eddies), andwidens as itdecelerates. If this all happens under the gate, and the upward bending causesreattachmentoftheshearlayertosomepartofthegateconstruction,thanthistoocanleadtoinstabilityifthisreattachmentisnotfixedinspacesomehow.
‐Athirdcauseofinstabilitytakesplacewhenthereattachmentisfixedtoonepositionbyaprotrudingelement(e.g.lipattrailingedge).Theregionbetweenseparationpointandreattachmentpointformsaclosedvolumeofrotatingflowwiththemovinggatebottomononesideandtheshear layer feedingenergy intothisregionontheotherside.Theenclosed area develops periodic pressure fluctuations which exhibit distinct, i.e.discrete excitation frequencies connected to howmany wavelengths of the pressurewavesareactiveinthegivenspace.
Adistinguishablefeaturecomparedto(ii)isthatamotionofthegatecanhaveinfluenceonthe cyclesof these flow instabilities and thereforeon the resultingperiodic forcing: activeinteraction.Inparticular,theexcitationfrequencycanadjustitselftogetclosertothemotion
35
frequency,the‘lock‐in’.InordertoanalysetheseflowinstabilitieswithCFD,theturbulence‐relatedfeaturesthatdefinetheseparationandtheshearlayermustbewellrepresented.(iv).Self‐excitationSelf‐excited or self‐induced vibrations are widely studied because they describe manyproblemsinengineeringandphysics.Theyarealsointerestingobjectsofstudyintheirownright, indynamicalsystemsanalysis(Verhulst,1996).Thisvibrationtype isdefinedbythedriving force originating from the displacement of the oscillating body itself (DenHartog,1956). That is, a self‐sustained system exists without the need for external forcing. Newenergyisfedintothesystembyanenergytransferfromthewaterflowtothegatemotion,which ismost easily described in the linearODE ofmotion (equation 2.13) by a negativedampingterm.Thisisclearlyanactiveinteractionthatcouldleadtoexponentiallyincreasingamplitudes. The general rule is that all forms of self‐excitation of gates are potentiallydangerousandshouldalwaysbeavoided.Threetheoriesofself‐excitationarediscernedforgates.GallopingEminated fromastudyofvibrationsof telephonewirescovered in ice,DenHartog(1956)explained that the observed low‐frequency oscillations could not be caused by vortexshedding.Instead,gallopingisadynamicinstability,linkedcloselywiththeobject’slay‐out,where the varying orientation of the incoming flow relative to themoving body gives anamplifying lift force in the same direction as this movement. The condition for whichdynamicinstabilityoccurs,asderivedbyDenHartog(1956)reads
0. (2.27)
That is, the change in the lift force coefficientCL as a functionof the angleα between theapparent or relative velocityUrel of the approach flow relative to thebody’s velocity is nolongerbalancedbythedragforcecomponentCD.Forarectangulargateprofileofavertical‐liftgate,anupanddownmovementmaycausecyclicvariationinthesuction(lift)force:inthedownward swing the regionbetween gate bottomboundary and shear layernarrows,local pressure decreases and downward suction increases and vice versa for the upwardmotion. To the extent that the time lag between excess lift andmotion gives rise to a netforceproportionaltoobjectvelocity,therewillbenegativedampingandself‐excitation.FluctuatingleakagegapThedynamicforcesonaplugvalve(orstopper)whenemptyingabathtubaredescribedbyKolkmanandVrijer(1977).Aspring‐mountedgateorvalveinitiallycoveringtheupstreamside of a culvert regulates the discharge from the upstream reservoir to the culvert. Theinertiaoftheflowingmassoffluidpastthegatecausesaliftforceonthegateproportionaltothevelocityatwhichthegateisraised–atasmalldistancefromtheopeningtotheculvert.SettingupODEs for thedischargevariation, the flow in theculvertand thepositionof thegateresults ina thirdorderODEwithphysicalcoefficientsandastabilitycriterion for thespringstiffness.
36
A generalisation step is needed to extend this theory from a gate closing off pipes andculvertsontheupstreamside,tovertical‐liftgatesinfreesurfaceflows.Thein‐lineinertiaisnowthoughttobecausedbyafluidvolumeinafictitiousstreamwisetubeoflengthD+(CLu+CLd)a,wheretheCiarelengthcoefficients.Figure 2.9 gives an impression of the effect. Under the assumption that the local headdifference at the gate consists of a constant global head difference plus a dynamic termcausedbythedynamicsofflowaccelerationanddecelerationrelatedtodischargevariationsdq/dt~da/dt~ ,itisfoundthattherehastobeanoscillatoryterminthesuction(lift)forcethat enables the changes of momentum of the imaginary tube. A lumped suction forcecoefficientwascomputedasafunctionofaStrouhal‐typedimensionlessnumberrelatedtothe gate opening and compared to experimental values. This theory is also discussed inKolkmanandJongeling(1996),buthashadlittlefollowing,despitetheappealingintuitionattheheartofit.FluctuatingdischargecoefficientAnother way of thinking about how the movement of the gate cuts off the discharge isthrough an oscillation of the discharge coefficient. This is for instance found in in‐flowvibrations of underflow gates with rounded bottom profiles. A moving point of flowseparationthencausesavaryingcontractionandsoavaryingdischargecoefficient,leadingtopressurefluctuationsacrossthegateinphasewiththevibrationvelocity.Theworkingofthismechanismand theprevious onedepends on the frequencyof themotion:when thisbecomestoohigh,theinertiaordischargecapacityeffectshavenotimetoinitiate.The last two theoriesarequite complicatingeffects specific togates.Theyhaveby farnotbeenstudiedasmuchasforinstancegallopingandvortexshedding.Allthreetheoriesin(iv)have incommon that the internaldriving forcedependson themotionamplitude.That is,comparedtothesimplefreemotionofequation2.13withFw=0thereexistsanextraforceF(y)proportionaltoyor ,whichoftenimpliesnon‐linearity.(v).UnstablefluidresonanceGlobalinstabilitiesofthefluidinwhichthegateislocatedcangiverisetogatevibrationwitha frequencydictatedby thenatural frequencyof the fluid reservoir.Examplesare seiches,standing waves of the free surface, standing compression waves. Incorporating thismechanismintheevaluationofgatestabilityrequiresabroaderviewonthehydrodynamicsthanfortheothermechanisms,whereanalysisofdetailedlocalflowwasneeded.Initiallysmallgatevibrationsmayalsobeamplifiedratherthaninitiatedbyfluidoscillations(‘self‐control’) inthereservoiradjacenttothestructure.Ifthecharacteristicfrequenciesofgateandfluidresonancecoincide,mutualenergyexchangecontributestofurtheramplitudegrowth.Thisphenomenonhasbeen recognised relatively recently. JongelingandKolkman(1996)providestudiesofreal‐lifecases.2.4.4 AddedcoefficientsThepresenceofthebodyinwatergivesaddedcoefficientsthat,indeed,havetobeaddedtothebasiccoefficientsofthelinearODEofequation2.13.Theaddedcoefficientsmw,kwandcwarekeytoquantifyingandunderstandingvibration(causes)ofbodiessubmergedinwater.Theyinfluencethenaturalfrequency,therestoringforcesandgovernenergyexchangewith
37
theflow.Kolkman(1976)establishedinsightfulconnectionsbetweentheaddedcoefficientsandthedimensionlessparametersinTable2.1.FollowingthecompactoverviewbyKolkmaninNovak(1984), theyaremerely introducedheretopreventrepetitionofalreadyexisting,apttreatments.AddedmassmwWhenabodyismovingthroughafluid,ithastopushsomeportionofthefluidoutofitsway.Themass of the fluid volume pushed away by the body (during oscillation) is called theaddedmass,writtenmw.Thisinertialeffectcausesthenaturalfrequencyofavibratingobjectinwatertobelowerthaninair.InFigure2.9thethreeinertialeffectsidentifiedforacross‐flowgatevibrationareillustrated.Thereisnostraightforwardanalyticalwayofquantifyingtheaddedmassingeneral.Onlyforverybasicobjectshapes inastagnant fluid,analytical formulasexist formw,undercertainassumptions.AnoscillatingstripoflengthLandwidthD,forexample,isthoughttomobiliseasemi‐cylinderoffluid,thusyielding 4⁄ 2 .BothpotentialflowtheoryandFEMhave been used to estimatemw for numerous applications. The value ofmw is in any caseaffectedbyconfinementsofthesurroundingfluiddomain:thefloorandwalls.Consequently,mwcandependongatepositiony,thusintroducinganon‐linearity.Otherfactorsthatmayormaynotcomeintoplayarewaveradiation(theproductionofsmallamplitudefree‐surfacewavesbythevibratingbody)and,relatedtothis,dependencyofmwonvibrationfrequency.
Figure2.9.Threeinertialeffectsconnectedtoaccelerationsofsolidandfluid.HydrodynamicrigiditykwAlsocalledaddedstiffnessorhydraulicstiffness,threeinstancesofkwareencountered.‐A floatingbodythat ispusheddownwardintothewaterexperiencesahigherupwardArchimedean force. Combining Archimedes with Hooke’s law, it is derived that adownward displacement ∆y of a semi‐submerged body gives an extra upward forceequalto ∆ ∆ ,whereAwisthehorizontalwetcrosssectionoftheobject.Itfollowsthatkwdoesnotdependongatepositiony and isproportional to theareaAwcuttingthewatersurface.
‐ A spring force is a restoring force; a wind vane rotating in the wind experiences apositiveaddedstiffnessforcepushingitbacktothestableequilibriumpositionparallel
mwy
my
ρCLay
gate mass
added mass fluctuating
discharge
38
to the ambient wind direction. When a lift force on a (wing) profile increasesproportional to the rotationwith respect to themean flow direction, it will give thesameeffect.
‐Athighgatefrequencythereisathirdkweffectcalledinstantaneousrigidityconnectedtothetheoryofthefluctuatingleakagegap.Supposeaculvertbetweentwobasinswithaheaddifferenceispartlyclosedoffbyanin‐flowgateonthedownstreamsideoftheculvert.Asuddenchangeofthegateposition∆xthatmakesthegapsmallerresultsinasuddenpressureincrease∆p(duetothesamedischargegoingthroughasmallergap),assumingthedischargefails toadapttothenewgateposition immediately.Theextrapressure acting on the gate surface Agate gives a force in phase with the suddendisplacement∆x and inworking in opposite direction. Thus, this is a restoring forcewithaddedrigiditykw=∆p∙Agate/∆x.Anequivalentanalysisforanupstreamculvertgateshowsthatinthiscasetherigiditykwisnegative.
AddeddampingcwPositive‐valued contributions to hydrodynamicdampingcw are radiationof kinetic energyfromthegateasfree‐surfacewaves(waveradiation),skinfrictionanddampingbydrag.Inthelastcaseitisnecessarytotaketheprojectionofthedragforceintheflowdirection,sayFy, and apply cw = / . This is a turbulent hydrodynamic damping proportional to thesquareofthevelocity.Negativevaluesofcwareinextricablylinkedtoself‐excitation.Howcwiscomputedisdifferentforeachofthethreeself‐excitationmechanisms.Intheend,itcomesdown to checkedwhether c + cw < 0 can occur,where c is the structural damping in themechanical system of gate and suspension. Thiswould lead to a dynamic instabilitywithincreasingamplitudes.Analogously,k +kw <0,withk the structural rigidity,wouldgiveastaticinstability.Thetermstructuraldampingisalsousedforananalyticalmodelknownashysteresis damping, the damping force then depends on displacement instead of velocity(e.g.Adhikari2000,Rijlaarsdam2005).Thismeaningisnotusedinthisthesis.Itiswisetoleave all options open regarding damping models; to name one more: Coulomb or drydampingisdescribedbyFdamp~sign( ).Sometimesenergyconsiderationscomeinhand.Theworkdonebythedampingforceonthebodyduringonecycleisequalto
, , ∙ (2.28)
overamotionperiodT.Thiscanquantifyadissipationofenergyoratransferofenergytothemotiony(t).It is noted that added mass and damping can cause pressures on the gate in a differentdirection than the vibration direction. The notorious example is the added mass of averticallyvibratingL‐shapedobject.This isabasicexampleofacouplingeffect; the forcesduetotheaddedmassinstillwaternowrequirea3x3coefficientmatrixrelatinghorizontalandverticalforcesandanin‐planemomenttoxandy‐translationandrotation.2.4.5 ConsequencesofgatevibrationsSmall irregularbackgroundvibrations,asdescribedby(i) in2.4.3,are impossible toavoidandarenotharmful.OtherFIVsmayalsohave innocentconsequences,but theproblem isthatwecannotarriveat this conclusionwithoutknowing theexact forcingcharacteristics,
39
assessingtheminstructuralanalyses,andcomparingcomputedstresseswithdesignvalues.Distinguishingthreefailurecategories,towhichofcoursearestcategoryofunknownfailuremodes has to be added, FIVs of gates will be called unsafe when one of the followingscenarioscomesintoeffect.‐ A (part of) the gate or adjacent structural elements is excited into a vibration withamplitudes exponentially increasing over time, until the point where this growthstabilizes due to extra internal or externalmechanical damping. At thatmoment, thesheermomentumandkineticenergyoftheoscillationcanbeenormous.Thisistypicalforvibrationsoftheself‐excitedtypeandshouldalwaysbeavoided.Itisprobablethatthestructuralelementsinvolvedinthisprocess(includinghinges,pivotsandsupports)werenotdesignedtowithstandthis.Particular for this typeof loading is theperiodicloadchange;combinedwithalongdurationthereisadangerforfatigue(accordingtoKolkmanandJongeling,1996).Failureofoneelementmaysetoffamaliciouschainofevents leadingto failureofotherelements. Inanextremecasereportedby Ishiietal.(1980),amaintrussfailsunderbucklingandthegatebodyasawholedisconnectsfromthestructureandiswashedawaydownstream.
‐ Gate vibrations hold up or hinder operation. As a result of vibrations, a gate can getstuck inonepositionorpartlybreak itselfbysmashing into thesill. It thenceases tofunctionasadischargecontroller.Ifnotimelyactionistaken,suchasinstallationofanemergencygate, thedischargecanbecomeuncontrollableduringasignificantamountof time. Apart from secondary inconveniences (e.g. shipping traffic), the immediatedanger is that local flowsexceeddesign limitsof thebedprotection.This isaseriousthreattothestabilityoftheoverallstructure.
‐Another typeof ‘failure’ comprises thepsychological impactofunexpectedvibrations.Hydraulic structures are seen as incomprehensibly strong and rigid structures thatwerebuilt toprotect landandcitizens.Anyobservationof vibrations, for instancebywhistling noises or small waves at the water surface, may lead to concern from themanagersofthestructure–withorwithoutgoodreason.
2.4.6 MythsaboutgatevibrationsThefactthatlinearvibrationtheoryandflowaroundagatearetaughtinelementaryratherthanadvanced courses in respectivelymechanical and civil engineeringmightmistakinglyleadtotheideathatgatevibrationisnotacomplicatedsubject.Muchonthecontrary,itisadeceptivelycomplextopic.Fivepopularmisconceptionsaregiven.(1).Additionofdampingcansolveanyvibrationissueofhydraulicgates.Adding damping is foremostly a (trivial) theoretical measure that, for large hydraulicstructures, israrelya feasibleandacceptablesolution;notonlybecauseasuitabledamperwouldhavetohavehugeproportions,butalsobecauseitwouldinhibitnormalopeningandclosingmovements.(2).Ifgatesvibrate,theyvibrateattheirnaturalfrequency.Thisisfalseasageneralstatement,foranumberofreasons.Firstofall,‘naturalfrequency’usuallyrefers totheundampednatural frequency inair.Thenatural frequency inwater isdifferentdue tomw andkw. Secondly, the gate canbe excitedby fluid resonance, inwhichcasethegatefrequencyadoptstheresonancefrequencyofthefluidbasin.
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(3).ProblemswithFIVinhydraulicstructuresarecausedbyturbulence.This is false. In practice, gates or other components of hydraulic structures are seldomexcited into problematic vibrations by turbulence alone. Non‐periodic noisy turbulentfluctuations in theapproach flowareas itwereoverpoweredby local flowaccelerationatthegateandthusactasarelativelysmallexternalload.Vorticesbythemselvesgiveapassiveforcingthatdoesnotgrowtotroublesomeproportions.(4).Intelligentgateoperationcanalwaysavoidvibrations.Of course it is wise not to choose critical (that is, often small) gate openings for longdischargescenarios.Asafirstsolutionapproachitisthereforegoodtoseeifdynamicloadscan be avoided or reduced by smart operation. This is investigated in Chapter 4 and 7.However,someexcitationscenariosarenoteasilyavoidedorrecognisedontime.(5).Adesigner thatknows thedangersofvibration‐sensitivegateprofilescaneasilypreventundesireddynamics.Tosomeextentthisistrue;theprobabilityofFIVisgreatlyreducedbyagoodgatedesign,preferablytestedinaphysicalmodel.Butthecombinedeffectofrealisedstructuraldetailsand local flow conditions can still causephenomena that arehard topredict andevaluatebefore construction. Consequently, vibration risks can never be ruled out completely.Moreover, there are several stringent constraints that usually claimmore attention in thedesignprocessthanFIVloads.
2.5 Generalisationandproblemsolvinginpractice
The singlemass‐spring schematisationwas chosenbecause it can suitably representmostfluid‐structure interaction mechanisms for gates. More complex and realisticschematisations can be constructed by adding multiple SDOF systems together, as iscommonplaceinstructuralmechanics.Thespecificchoiceforaverticallymovinggateobjectisontheonehandmotivatedbythefactthatvertical‐liftgateswithunderflowconstitutealargeclassofgates,andontheotherhandbythefactthatthisconfigurationgivesthemoststraightforward and generalisable description that can be adapted for application in thevibrationanalysesofothergate types.Thismightseemstrange.Forestimatingdischargesand hydrodynamic forces other than those caused by waves, ice and vibrations, differentgate types have to be treated separately. Indeed, textbooks such as Naudascher (1991)enumerate gates of many shapes and sizes and contain various, different formulae forestimatingtheforcesonthegateandthesupportforces.However,forFIVanalysisthereisoftenlittlesenseinrepeatingtheanalyticaldiscussionsofSections2.4.3and2.4.4forothergate types, for instance for a rotating tainter valve (see Figure 2.6), because there issignificantoverlapintheconditionsunderwhichvibrationsariseandthereforeprincipallythe same excitationmechanisms (Figure 2.7) are distinguished for different gate types. Inother words, the single mass oscillator submerged in a flow is a fairly universalrepresentation. Sometimes one encounters SDOF schematisations for theoretical analysisbasedonagatetypeotherthanthevertical‐liftgate,however;forexampleKolkman(1976)inoneofhisappendicesstudiesareversedtaintervalveinaculvert.Despiteadifferent(lesstrivial)equilibriumofstaticforces,thevibrationmechanismsofthisgatearesimilartothatof the vertical gate. More complex cases involve continuous or distributed masses (see
41
NaudascherandRockwell,1994),amongwhichareplates,beamsandshells.Becauseofthepresenceofwater,theoreticaldynamicresponseanalysesofsuchelementsquicklybecomeimpracticallyhard.Themainlimitationoftheverticallysuspendedgateschematisationisthatitonlyconsiderscross‐flowvibrations.In‐flowvibrations,inFigure2.1thesewouldbehorizontal,havebeenandare treatedseparately inacademic literature (e.g. Jongeling,1988).Asmallnumberofexcitationmechanisms (not part of this thesis) is thought to exist exclusively for in‐flowvibrations, but it cannot be denied that in‐flow and cross‐flow vibrations share manyfeatures.When a gate body experiences bothdegrees of freedom (cross‐flowand in‐flow)simultaneously, the situation is fundamentally more complicated. Acceleration in onedirectionmay ignite acceleration in the other due to coupling effects stemming from theaddedmasses.ItismentionedbyKolkmanandJongeling(1996),withreferencetoastudyofIshiiandKnisely(1992),thataratioofeigenfrequenciesofbothdirectionsclosetoonehastobeavoided.The analyticalmetaphor of the gate as a spring‐mounteddiscrete body forms the startingpoint of practical FIV studies of all kinds of gate types. The engineering practice basicallyrequiresacombinationofthisfoundationandadhocdetectivework.Ofcourseknowledgeabout how different gate types work andwhat hydrostatic forces they experience is alsoneeded.Butfromtheoryit is impossibleto linkonegatetypetoonevibrationmechanism;everythingthatwastreatedinSections2.3and2.4.1‐2.4.5ispotentiallyapplicabletoallgatetypes.Afewexamplesoftypicalrecurringphenomenaencounteredinpractice:‐dischargevariationsduetolocalleakagebetweengateandsill,‐ periodic pinching of slits (between gate and side support or between two adjacentgates),
‐lackofaerationonthedownstreamsideofgateswithoverflowdischarge;thiscangiveperiodic variations of the enclosed air volume, leading to pressure variations on thegate.
In case of suspicion of gate vibrations, the curing procedure needs to at least look atquestionslike‐Whataretheleaststiffbendingdirections?‐Whereisroomformovement(sideways,rotational,etc.)?‐Aretherepivotsthatallowacertainextrafreedomofmovement?‐Isthereanasymmetryof(leakage)flows?‐Dotheshapeandmaterialofthesealingsallowflowinstabilitiesorunwanteddischargevariations?
Thisresultsinanoverviewofthemostlikelyeigenmodes.Apartfrompossibleoscillationsofthe gate as a whole, it is necessary to sсrutinise the elastic properties of all structuralelements(especiallythosewithrelativelylowstiffness)thatcomeincontactwiththemainflow, e.g. plates near gate edges, supporting trusses, etc. Additionally, for each potentialvibrationelement,ithastobecheckedwhichofthemechanismsmayleadtoexcitation.Allthese issues are commented on and illustrated by case studies in Kolkman and Jongeling
42
(1996).Thesameauthorsnotethatifafieldstudyresultsinacorrectdiagnosisofvibrationsources,theimplementationofacountermeasureisoftensimpleandinexpensive.Experience clearly plays a large role, and proper judgements require the ability todistinguishmajorissuesfromsecondaryissues,butthisisnosubstituteforsoundreasoningandcomputationbasedontheratherwidetheoreticalbackgroundsketchedinthischapter.In their concluding remarks,Naudascher andRockwell (1994) advocate theproperuseofthe excitation theories: “Had FIVs been discussed exclusively in terms of affected systemsand structures, the reader would have been misled into thinking that by eliminating thevarious specific causes for excitation, it would be possible to safeguard similar systemsagainstvibrationproblems.WewishtoreemphasizethatinordertocomeclosetothegoalofascertainingallpossibleoriginsofFIVs,onemust,first,thoroughlyidentifyallthevariousbodyoscillatorsandfluidoscillators(…);and,second,onemustdiscoverallpossiblesourcesof parametric excitation as well as extraneously, instability‐, and movement‐inducedexcitations that might affect these body and fluid oscillators, either individually or incombination,forthegivenoperatingconditions.”Andfinally:“Engineersareadvised(…)touse the examples [from preceding projects] merely as a means of training themselves inrecognizing the basic elements of FIVs (...). Simpler roads or shortcuts to satisfactoryengineering solutions do not seem to exist in this complex field of fluid‐structureinteraction.”
2.6 Conclusionsfromthephysicaldescription
RecappingtheanalyticalmodelparadigmsofSections2.2‐2.4,plusthediscussioninSection2.5,anumberofconclusionsaredrawnwithrelevancetothecomputationaleffortsahead.‐Theaddedcoefficientsmwandcwcentraltoquantificationofdynamicexcitationsarenotreadilyderivedanalyticallyforanoscillatingobjectinaflowingfluid.
‐ The traditional engineering method of estimatingmw and the steady hydrodynamicforces on a gate, namely by potential flow theory, is too limited to have generalcomputationalpower.Itworksonlyforinviscid,rotation‐freeflowandthusfailstodealwithseparatedflowandanytypeofflowwhere(periodicornon‐periodic)fluctuationsdominate.
‐ Self‐excitation is typically a non‐linear phenomenon. Therefore, all linearmodels fallshortofafulldescription.
‐ Applying analytically derived equations as the sole foundation of numerical FIVpredictionsistroublesome,becauseofthecomplexinteractionofthefluid‐solidprocessand the essential role of empirical factors (e.g. discharge coefficient, lift forcecoefficient).
‐Thusfarinliteratureithasbeentherigourofphysics‐basedanalysisthatisusedintheidentification of vibration causes and in the search for suitable remedies. But theapparentorderlydinstinctionsbetweendifferentexcitationtypesandbetweendifferentadded coefficients are in practice much less obvious. There can be –and often are–combinationsofthemechanisms.Moreover,whenmorethanoned.o.f.comesintoplaythe number of possible physical descriptions explodes and the intangibility tends tomakeexplanationsofoscillationoriginsmoresubjective.
43
‐Realisingthatagateispartofacompoundstructurewithinnumerousstructuralparts,some of which are influencing the flow around the structures, makes matters evenworseintermsofpredictingwhetherFIVswilloccur.
Inconclusion,newmethodsforanalysisandcomputationsarewelcomed.Real‐lifeproblemsolvingwouldbenefit inparticular fromcomputational techniques that establishor refuteconnectionsbetweendominanteigenmodesandexcitationmechanisms.
44
45
3 Physics‐basednumericalmodelling
3.1 Introduction
Supposewearefacedwiththeproblemoffindingthegateresponsesignalfromasetofgivenflow conditions, acting as input, and a physical model for performing experiments is notavailable. Physics‐based numerical simulationswith the purpose of system analysis is thetraditionalrouteforsolvingthisproblem.Fromastructuraldynamicspointofviewthisisanexampleofadirectmethodinthetime‐domain.At the start of applying a numerical model, the main question is not which high‐level,continuous‐formdifferentialequationsdescribe thephysics.For fluid‐structure interaction(FSI),wehavetheNavier‐Stokesfortheflowandstress‐strainrelationsforthestructure–ingeneral it isassumed that suchabasealreadyexists,derivedviaFigure3 in section1.2.1.The questions are rather what choices are sensible in the discretization and numericalsolution of those equations andwhich boundary conditions are appropriate. In the inter‐pretationof themodeloutcomes, importantquestionsarehow to judgemodeleffectsandscaleeffectsandhencewhatthephysicalparameterrangesarewhereextrapolationisvalid.Section 3.2 gives a framework for the application of computational fluid dynamics (CFD)models.ThediscussionispurposelybiasedtowardsFSIanalysis.Section3.3thengoesintoafewspecificaspectsofFSI.Althoughputtingadetailedgeometryof thebarrier ina computermodel isdoable (albeittime‐consuming),themodellinggoalinchapters3‐6isnottolookatthestructuralresponseside in detail (i.e. by including trusses or beams and computing all stresses, etc.). This isbecausetheassumptionofarigidbodywithamass‐springsuspensionisaveryreasonableand generalisable analogy for true structural flow‐induced responses, while manysimplifyingassumptionsforthemodellingofthefluidflowaredisastrousforcapturingtheflowfieldandthereforetheflowloadonthestructure.
3.2 ApplyingCFDforproblemsolvinginhydraulicengineering1
It is remarkable how different choices in numericalmodel set‐up and use can shape andaffect the way problems are solved in hydrodynamics. Some philosophy about applyingmodelsisnecessaryinordertogetresultsthatanswertherightquestionandthatpertaintothe realworld.When talking about solving a real‐life physical problem, there is no singlepredefined algorithm that fits the job exactly. The result, after careful analysis andpreparation, ismore likely tobeacombinationofmodelling tools.Foreaseofspeech, thiswill still be referred to as a (numerical or computational) model. Figure 3.1 presents aframework aimed at preparing a model –or coming to an appropriate model choice– forsolving a problem in hydraulic engineering. This diagram is walked through step by stepfromtoptobottom.
1TheframeworkandgeneralpartofthediscussioninSection3.2isinspiredbydiscussionsaboutCFDuse,researchanddevelopmentatDeltares,mostlywithMartBorsboom.
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Figure3.1.ApplicationframeworkforCFDmodels.ProblemstatementIt starts with the identification of a physical problem. There is usually one question thatsumsuptheproblem.Iftherearetoomanyquestionsneededtopinpointtheproblem,thisisahintthatonemodelmaynotbeenough.Forthegate,threerelevantquestionsare“Whatarethemaximumdynamicforcesexperiencedbythegate?”,“Givenasetofflowconditionsandgatepositions,will thegate start vibratingornot?” and “Whichexcitationmechanismcanbeexpectedtocomeabout?”.PhysicalphenomenaInthefirststep,thetaskistocompilealistofthephysicalphenomenathatplayaroleintheproblem. Five classes of phenomena are mentioned: the influence of the free surface,turbulence,densityeffects,morphologyandmovingobjects.Forthevibratinggateproblem,density effects and sediment transport arenot important, but the other threephenomenaare.Theknowledgeneeded for judging the importanceof these three aspects comes fromChapter2.Thisisalsothemomenttothinkaboutscales.Whenthefluiddynamicstakeplaceinasmallvolumeand thesmallest flow lengthscaleshavecomparable sizes inall threedimensions,they can be called ‘detail hydrodynamics’. Then turbulence is a factor and the flow is bydefinitionnon‐hydrostatic,whichmeansthatthepressuredoesnotvaryproportionallywithdepthandthestreamlinesarenot(approximately)paralleleverywhereinthedomain.Flowaroundastructuredefinitelyfilesunderdetailhydrodynamics,butthisdoesnotmeanthatitisnotalsoaffectedbyconditionsawayfromit–whicharedeterminedbysystemsoflargerscale(callednear‐fieldandfarfield).Chapter4goesintothis.
validation
problem statement
physical phenomena
modelling requirements
computational model
evaluation
47
ModellingrequirementsThe now known physical phenomena can be modelled in various ways. It should bedetermined, in order to address the problem, which physical aspects should explicitly berepresented by the model and to what level of detail. It is said that quantities that arecomputedexplicitly(fromfundamentalprinciples)asafunctionoftimeare‘simulated’,andquantities that are not, but do play a role in the problem and are part of the system, are‘modelled’. Tomodel rather than simulate aphysical quantity involves a certaindegreeofparametrization, i.e. representationbymeansof lumping, averagingorestimatingof somekind.PracticallyallCFDandFSImodelscombinesimulationandparametrization.Browsingthroughthelistofrelevantphysicalphenomena,itshouldbedeterminedwhichneedtobesimulatedandwhichcanbemodelledinasimplifiedway.Anumberofadditionaldecisionsarenecessarytoo.Forourproblemthelistisasfollows.(i).TimedependencyThe flowaroundastaticgate isquasi‐stationary(Section2.2)andcouldbemodelledwithsomedegreeofaccuracybysolvingsteady‐stateflowequations.Theinclusionofamovablegate implies that the problem becomes fully transient (i.e. time‐dependent). In fact, themodelhastosimulatethegatemovementinordertodojusticetothefluid‐solidinteraction.(ii).Whatisthedegreeofnon‐hydrostaticity?Theverticalcontractionoftheunderflowandtherecirculationbehindthegatecallforanon‐hydrostatic approach.Any simplificationof the equations concerning vertical acceleration,e.g.asisthecaseintheshallow‐waterequations,willnotsuffice.Whetherthemodelhastobethree‐dimensionalisanothermatter.Atwo‐dimensionalgatesection(uniforminwidth)provides a good model (Sections 2.2‐2.4), but the turbulent kinetic energy may beunderestimated, because v′ 0 in themodel. The added value of a 3D flow simulationmostlikelydoesnotoutweightheenormousextracomputationalcosts.(iii).Howimportantisthefreesurface?Thisdependsonthedischargeconditionsthatwewanttoexamine.Moreover,theexpectedfrequencyofthegatemotiondeterminesifpotentialwaveradiationisperceptiblenearthesubmerged zone of excitation at the gate bottom. If the problem is restricted to fullysubmerged flow with negligence of wave radiation, then a simulation of free‐surfacefluctuations seems unnecessary. As the downstream Froude number increases, thissimplificationbecomeslessandlesssuitable.(iv).Towhichlevelofdetailshouldturbulenteffectsbesimulated?Ithasbeendiscussedthatturbulence is inherentlypartofthe flowaroundthegate.Atthesametime,theexcitationmechanismsmostlikelytocausehigh‐amplitudevibrationsarenotdirectlylinkedtoturbulence.Theturbulenteddiesatthesmallest(Kolmogorov)scalesintheviscousrangeoftheturbulencespectrumaresurelyimportantinthedecelerationzoneawayfrom the gate and play a role in hydraulic damping due to the gate movement. But theexcitation itself is presumably affected more strongly by the relatively large‐scalefluctuations(intheinertialrangeofthespectrum)originatingrightawayfromthoseregionsof the free shear layer with high stresses. Therefore, a parameterised modelling ofturbulence is suggested that respects this.Thedifficultyof thispoint lies in the fact thata
48
pre‐modelling dismissal of certain excitationmechanisms to somedegree undermines thegoalofthemodelling:analysisoftheprocess.(v).MovingobjectIthasalsoalreadybeenestablishedthatthegatewillactasapartlysubmergedoscillatingobjectwithoned.o.f..Therequirementofsimulatingamovingobjectstronglyinfluencesthemodel set‐up with respect to mesh (see Section 3.3.3) and time discretization. ExpectedfrequenciesareestimatedfromtheaddedmassofarectangularobjectnearawallandtheArchimedean‐type hydraulic rigidity in the natural frequency formula; these indicate theminimumtimetimestepsfromaphysicsperspective.(vi).Whatkindofboundaryconditionsarerequired?Thefloorboundaryoftheflowisbestmodelledasaroughwall.Theupperboundaryoftheflowdomainisthefreesurface,whichwasalreadydiscussed.Inflowmodelswithmorethanone computational layer in the vertical, it is common to place a hydrostatic pressureboundary downstream and a velocity profile upstream. This is for good reason; there aremany ill‐definedordownrightunstableboundary combinations (StellingandBooij, 1999).Theinclusionofwaveswouldmakeespeciallytheoutflowboundaryradicallymorecomplex(e.g.Wellens,2012).ComputationalmodelThe step from modelling requirements to computational model starts with aninventarisationofavailablecomputational tools.Hereweare lookingforaCFDmodel thatcan handle detail hydrodynamics and that can be extended to include the interactionbetween the flow and a moving object. Now we become concerned with numericalrequirements such as discretisation schemes, grid type, solvers, etcetera. Trivialrequirementssuchas“themodelneedstobetime‐efficient,accurateandflexible”allfallinthecategoryoffinitecomputationalcapacityandavailabletime.Insummary,theeverlastingtrade‐offbetweenlevelofdetailofmodelledphysicsandsimulationtime(timesnumberofrequiredruns)needstobeidentifiedinanearlystage.Itisgoodtothinkabouthowtospendyour limited time as amodel user. In CFD, preparing a run (‘pre’) and processingmodelresults (‘post’) canalso takea lotof time.Howmuchdoyouwant todobyhand (manualcoding)andhowmuchcanbedonewithexistingcodesorsoftwarepackages.EvaluationandvalidationAfter running the numerical model the results are visualised and evaluated. In the bestscenario this leads to usable answers to the stated questions and recommendations forsolving the (engineering) problem. The first series of runs never leads to the desiredsolution,however.There iscritical feedbacktocheckallpreviouschoices– thisprocessofbuildingconfidenceinthemodeliscalled‘validation’.ValidationworksonallstepsofFigure3.1andmustbeconsideredtobeaniterativeprocess,aloop,ratherthanaone‐offactionoramodel property, in the spirit of Dee (1993). Validation actions are concerned with thequestioniftheabovestepsaretakenintherightdirection,andnotiftheyareimplementedcorrectly;thisiscalled‘verification’.Furthermore,a‘calibration’isasetof(additional)stepsthat make a physics‐based computational model ready for a specific application. This isusuallydonewithanotherdata set (notusedbefore invalidation). So, as a resultof thesedefinitions,itisstrangetospeakabouta“validatedmodel”,butitispossibletospeakabouta
49
“calibratedmodel”. A computationalmodel that has been tailor‐made for solving a statedproblemiscalleda‘modelapplication’.Inconclusion, thecentral ideaof thesesteps is tomoldthemodel intoaworkable tool foranswering the initialquestions, andnot theotherwayaround (i.e. to search forquestionsthatcanbeansweredbyyourmodel).Aninterestingcorollaryisthatabetterdescriptionofthe underlying physics or technical improvements to the algorithm may or may notcontributetobetteranswers.Forthegatevibrationproblem,thecrucialchoicesrelatetoabovemodellingrequirements(iii),(iv)and(v).Thesearediscussedfurtherinthenextsection.Fromtheonsetourprobleminvolvedanalysisbysimulationandwasnotapredictionproblem(Section1.2),butwiththeinclusion of data for validation and calibration this distinction is starting to blur. If pastoutputdataisavailableanddirectlyusedintheprocessofcomputingfutureoutputvalues,thenthemodelisusedforprediction.However,inFSIcontext,itcouldwellbethecasethatthe validation data relates to the flow only – implying that in a strict sense the flow ispredictedandthestructuralresponseissimulated.
3.3 SettingupaFIVmodel
3.3.1 FiniteelementmethodThe finite element method (FEM) is a numerical method for solving boundary valueproblems.Itisbasedonaspatialdiscretizationinacomputationalgridor‘mesh’.Therearetwoothergrid‐basedmethodsinCFD:finitedifferencemethod(FDM)and,themostwidelyused, finitevolumemethod(FVM).Differencesbetween theseapproachesarediscussed inVanKanetal.(2008).TheystillhaveplentyofcomputationalfeaturesincommoncomparedtoCFDmethodsthatarenotgrid‐based.Forinstance,asharedissueistheimportanceofgridsizeandgridtype,sincewearedealingwithaconvectionproblem,seeagainVanKanetal.(2008). Another typical aspect is that a large system of equations is being solved. Theindication‘large’referstothefactthatstandardsolveralgorithmswillrunintotroublewhenfacedwithsystemsofthissize.Ameasureforthecomputationalloadisthegridsize,i.e.thenumbersofcellsandnodes,timesthenumberofcomputedtimesteps.TheFEMisalsoaclassicaltoolforsolvingeigenvalueproblemsfortheanalysisofstructuresand elastic solids (Hughes, 2000). Eigenvalue problems (or ‘eigenproblems’) describe allsorts of frequency analyses such as the problem of determining natural frequencies andmodeshapesoffreevibrationsofstructuralelements.Modalanalysis,whichinvolvessolvingeigenproblems, is usually performed in the design stage of structures, in order to makesafetyassessments.Theabilitytosolvedeformationsandstressesofasolid(withacomplexgeometry)andthefluidflowmakestheFEMasuitableapproachforFSIproblems.3.3.2 TurbulencemodellingDirect numerical simulation (DNS) is a very costly technique that simulates all turbulentscales.Allothermethodsapplyparametrizationof the turbulentdynamics tocutdownoncosts. Large eddy simulation (LES) divides the turbulent energy spectrum up in aparameterizedpartforthesmallerscalesandasimulatedpartforthelargerscales,butthisisstillacomputationallyintensiveapproach.ApplicationofDNSforreal‐lifecasesathighRe
50
bringsgreatnumericalchallengesandisthereforeveryrare.LESismoreaccessible,butstillfar from a standard option for CFD applications. The household approach for real‐lifeapplications is to use a turbulent model. In this case the turbulence is approximated byaveraging the Navier‐Stokes equations after substitution of the Reynolds decomposition(Section2.2)andtosolveanextrasetofequationstoapproximatetheReynoldsstresstermthatdescribestheturbulence(seee.g.ZienkiewiczandTaylor,2000).Forthepresentapplicationthreeargumentsaregivenwhythis lastapproachwillbeused.Firstofall,asarguedintheprevioussection,thereisreasontothinkthatsimulationofsmall‐scale turbulence isnotessential forrepresenting themost importantexcitations.Secondly,combining LESwith amoving object and possibly amoving free surfacewould become avery costly and experimental endeavour – not in linewith the aims of this study. A thirdreasonisthatit isnotsensibletocomputesmallturbulencescalesforconcreteapplicationpurposesif thereisnoflowmeasurementdataavailableatall– letalonedatathatcoveracomparablelevelofdetail.ThemainflowequationswillbegiveninSection6.1.Thetwo‐equationk‐εmodelisusedtomodel the Reynolds stresses, this is called the ‘closure’. This turbulencemodel works byestimatingtheturbulent(eddy)viscosityνTusingtheformula
, (3.1)
where cμ is a fixedmodel constant andk, the turbulence kinetic energy (TKE), and ε, theturbulentdissipation,arebothsolvedfromtheirowntransportequation.DetailsarefoundinRodi(1993),agoodreferencefortheanalyticalbackgroundanduseofturbulentmodelsinhydrodynamics.Thetwoextravariablesoftheturbulentmodelneedinitialandboundaryvalues.Moreover,theirtransportequationsarenotvalidinthevicinityofwalls.Thisiswhyawallfunctionisnecessarytoestimatethevelocitiesandturbulentproductionanddissipationnearwalls.3.3.3 ThearbitraryLagrangian‐EulerianmeshModellingamovingobjectinteractingwithfluidForagrid‐basedcomputationoffluid‐structureinteractionwehavetoensurethatthebodyisable tomove freely throughthefluiddomainandthat itexperiencesstaticanddynamicwaterforces.Tocapturetheactiveinteraction,thefeedbackeffectofachangedpositionofthe body should also be included. This implies for the numerics that in addition todescriptionsof the fluidandsoliddomainandtheiroutsideboundaries, conditions for thefluid‐solid interface are necessary. These consist of dynamic and kinematic conditions,expressedinthreeparts(Walletal.,2006):‐forceequilibirum,i.e.equilityofstressvectors: ∙ ∙ ,‐nomassflowthroughtheinterface,i.e.equalityofnormalvelocities: ∙ ∙ ,‐ thefluidviscosityalsogivestangentialcomponents,resultinginthe ‘no‐slip’condition
.
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Heretheboldsymbolsrepresentvectors,nisthenormalvector,uistheflowvelocityandyisthesolid’sdisplacement.Iftheseconditionsaremetprecisely,thisimpliesaconservationofmass,momentumandenergyattheinterface(Walletal.,2006).Theresultingsystemofequationscanbesolvedindifferentways.Thesehavetodowithaprincipaldilemma:mostflowmodelsuseaEulerianapproachandmostsolidmodelsuseaLagrangianapproach,andbothdomainsneedinformationfromeachotherattheinterfacetocomputethenextstep(DunneandRannacher,2006).Thevariousclassesofsolveroptionsare shown in Figure 3.2. The first option is a separation of the two domains. This is the‘partitioned’or‘segregated’approach.Thedomainsaresolvedseparately,i.e.inseries,usingan initial guess for the solid interface. In ‘iterative’ (or ‘iteratively staggered’) solvingalgorithms, in each time step, the serial steps are repeated until the solutions for bothdomainsconvergewithrespecttotheinterfaceconditions.Asimplerandfasteroptionisthe‘sequential’ (or ‘sequentially staggered’) scheme where you do the same with only oneupdate step for the coupling information. That is, the initial prediction of the soliddisplacementisreplacedonceforeachtimestepafterthefluidandstructuraldomainshavebeencomputed.
Figure3.2.Computationalapproaches for fluid‐structure interfaces,basedonterminologybyWalletal.(2006)andDunneandRannacher(2006).Inadifferentsetting,theequationsofbothdomainsaresolvedtogetherasonesystem;thisis the ‘fully coupled’ or ‘monolithic’ method. Two reasons why not to use fully coupledsolving is that this prevents the use of dedicated solvers for eachof the domains and thecomputationaldifficultiesarisingfromthelargersizeofthesystemmatrix.Anadvantageisof course the strong coupling typewhich –once solved– leads to a robust solution of theinterfaceposition.OnewaytodealwiththeadaptationofthecomputationalmeshtonewinterfacepositionsateachtimestepistointroduceamappingφbetweentheEulerianfluiddomain,withfixedorspatial coordinates,andadeformingreferencesystemχ (followingnotationbyWalletal.,2006). All physical quantities find their actual position in the physical space x from the
sequential (staggered)
solving the discrete equations
fully coupled (monolithic)
partitioned (segregated)
iterative(staggered)
52
transformationx=φ(χ,t)beforetheyareactuallycomputedinthefixeddomain.Figure3.3illustratesthis.
Figure3.3.MappingusedinALEmesh,diagramafterWalletal.(2006).This deformation mapping has to be determined anew on each time step as part of thecomputations. Analytically, the transformation is achieved by inserting a time‐dependentJacobian of the mapping, Jt = det( ⁄ ), into the Navier‐Stokes equation. The methodsketchedhereiscalledthearbitraryLagrangian‐Eulerianmethod(ALE). Itbasicallyadaptsthe fluid domain to changing boundaries and lets the structural domain keep its favoriteLagrangian form. Specific background in differentmodelling contexts is found in FerzigerandPerić(2002)andDoneaetal.(2004).Theprecedingdiscussiononlytellshowthemeshdealswithchangingdomainboundaries.The interior nodes naturally also need to change their positions in time following theseboundary changes and internal deformations (in the structuraldomain). This is organisedviaanextrasetofmeshsmoothingequations.TheALEmeshadaptationonlyworksifthenodeconnectivitiesremainintactandgridcellsdonotturninsideout.Thisputslimitsonwhatkindofsqueezingandstretchingisallowed,that is, on the (rate of)mesh deformation. Oneway to keep going when these limits areexceededistoremeshthewholedomain.Thiscomesatahighcostwhentherearefinemeshregions, which there typically will be in the fluid domain. Of course, even without meshdeformations themeshcancontainsubstandardelements, e.g. tight cell anglesor stronglyelongatedcells,sothatthemodelerneedstokeepaneyeonmeshqualitybeforeandduringthesimulation.ThefreesurfaceDynamicforcescausedbythepresenceofwateraroundastructurefallintotwocategories:wavesandflow(sometimescalled ‘current’).Thesearetwoprofoundlydifferentclassesofphysical phenomena and therefore lead to drastically different modelling requirements.Simulationof (long‐period)oceanwavesor (e.g. ship‐induced)short‐periodwaves in timeandspaceisalreadyacomputation‐intensivetask,socombiningthiswiththesimulationoffloweffectsinthesamemodelatrealisticallyhighReisnearlyalwaysavoided.Forsimulationofthefreesurfacenearastructurethereareseveraloptions.TheALEmethodissometimescalledaninterfacetrackingmethod,asopposedtointerfacecapturingmethods.
χ2
χ1 x1
x2
χ1
φ(χ,t) χ2
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An example of the latter is the volumeof fluidmethod (VOF). It couldbedescribed as anEulerian‐Eulerianformat,withafixedCartesianbackgroundgridandaspecialfunctionthatdeterminesthedensityratioofthetwofluidsinsideeachcell.Thismethodcanbeappliedforfluid density problems (fresh‐salt) and free‐surface problems (air‐water). The packageCOMFLOW is an example of a VOF model (Veltman et al., 2007). It is used in hydraulicengineering to simulate localwaveeffects fromwhich theheightsofwavesslamming intooffshore structures such as wind turbines can be computed. See Wellens (2012) forbackgroundonVOFandnumericaltestswithCOMFLOW.Otherinterfacecapturingmethodsare the ‘level set’ and the ‘phase field’methods.Relatively recently otherCFD approachessuch as the smooth particle hydraulics (SPH) method and the lattice Boltzmann method(LBM)havealsogainedattention.IfhighFroudenumbersdownstreamofthegate,inparticularhydraulicjumps,areexcludedfrom the range of analysed conditions, it makes sense not to use interface capturingmethods,becausethesearemoredifficulttocombinewithturbulentflow.Instead,theALEmeshcanbeused for trackingmild flow‐inducedsurfacedisruptions.The freesurface isaspecialboundaryinthesensethatthepressureandtangentialforceoughttobezerothereandthatitcorrespondstotheouteredgeofthematerialfluidparticlesforeverytimestep(ZienkiewiczandTaylor,2000). Inessencethesameinterfaceconditionsapplyas listedatthestartofthissection,butwithouttheinfluenceofasolid.ItisdecidedtoexploretheALEmeshinthisstudyformodellingbothforegoingphenomena:the moving gate and the free surface curvature behind the gate. Table 3.1 contains anoverviewofthetwomodels.Table3.1.Characteristicsofthetwonumericalphysics‐basedmodels.
chapter
time
dependency
gatemotion
turbulence
modelling
freesurface
validation
data
modelI 4transientwithsteadypre‐run
fixedk‐ε
modelALEmesh
Nago(1978,1983)
modelII 6 transient1d.o.f.(vertical
translation)
k‐εmodel
rigidlidexperimentaldata
gatemotion(Chapter5)
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4 Multi‐scalemodelfordischargecontrolandflowimpact2
4.1 Introduction
Thischapterexploresthe incorporationofaphysics‐basednumericalmodel forsimulatingthe local flow around the gate of a hydraulic structure in hydrodynamic flowmodels of alargerscale.Thelocalflowiscapturedbothbyparametrizationandsimulation.Thepurposeof this modelling effort is twofold: (i) to investigate whether this approach can givereasonable estimates of gate discharges usable for discharge control, and (ii) use modelresultsforjudgingtheimpactoftheflowonthestructure.Thedesignationmulti‐scaleinthechapter titlerefers tothe fact that the local flowmodelruns insideamodeloperatingatadifferentphysicalscale.Noexplicitlinksaremadewithresearchoncomputationalaspectsofmulti‐scalemodelling.Barrieroperationiscommonlybasedonwaterlevelpredictionsfromsystem‐scalefar‐fieldflowmodels containing river sections and sometimes coastal regions. The procedures areaimedatfulfillingthemainfunctionofthestructure:foraweirinariverthisistomaintaintheupstreamwaterlevel,foradischargesluicethisistotransferriverwaterouttotheseawhilekeepingasafe inland level.Present‐dayhydraulicstructureshavevarioussecondaryfunctions,suchasprovidingfavourableecologicalconditions,forwhichusuallynonumericalaidsareavailableindailyoperation.Abetterpredictionoftheflownearstructureswouldbebeneficialtodurableperformanceofallbarriertasks.Properdesignstudiespayattentiontoallfunctionsofastructureandassesstheimpactofallrelevantflowfeatures.However,operationalconstraintschangeintimefornaturalreasons(e.g. sea‐level rise) or political reasons (e.g. “Kierbesluit Haringvlietsluizen”, seeRijkswaterstaat, 2004). In addition, sometimes the design criteria that were originallyappliedcannotberetrieved,yieldinguncertaintyaboutsafetylevelsandallowablelimitsofgatesettingsinthepresent.Thereareseveralaspectsofcontemporarybarriermanagementforwhichaninformedviewondischargeandflowpropertiesaroundgates isessential.Apart fromdynamicgate loads(vibrations), the prediction of bed material stability and scour, including local erosion(Hoffmans and Pilarczyk, 1995; Azamattula, 2012) as well as large‐scale morphologicalchanges of surrounding bathymetry (Nam et al., 2011) greatly depend on the flow. Theseaspectsrelateto(long‐term)safetyofagingstructures.Additionally,ecologicalissuessuchasfish migration, salt water intrusion and mobile fauna are also linked with local flowcharacteristics(Martinetal.,2005).Thepossibleimpactofflowaroundstructuresonnearby
2Thischapterusestextandcontentfromthepaper“Free‐surfaceflowsimulationsfordischarge‐basedoperationofhydraulicstructuregates”byC.D.Erdbrink,V.V.Krzhizhanovskaya,P.M.A.Sloot,publishedinJanuary2014intheJournalofHydroinformatics,Vol.16,No.1,pp.189‐206,http://dx.doi.org/10.2166/hydro.2013.215.
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shipping traffic is another feature. Above considerations motivate quantification of flowaroundahydraulicstructure.There exist several non‐hydrostatic CFD sotware options for simulating detailhydrodynamics (e.g. ANSYS‐CFX, OpenFOAM, COMSOL, STAR‐CD). Bollaert et al. (2012)employnumericalmodellingtoassesstheinfluenceofgateusageontheformationofplungepool scour of a hydropower dam. Scheffermann and Stockstill (2009) made a transientsimulationof a closing gate in a culvert of a navigation lockwith STAR‐CDand comparedcalculated pressures with physical model data. Numerous other numerical studies havelooked into sluice gate flow (Akoz et al. 2009;Kim2007;Khan et al. 2005), but transientsimulationsofboth turbulent levelsand the freesurface remainchallenging.Furthermore,estimating the discharge overweirs and over or under gates is not trivial.Newdischargeequationsarestillbeingintroduced, fromdata‐drivennumericalmodelling(KhorchaniandBlanpain,2005)andfromphysicalexperiments(Habibzadehetal.,2011).System‐scalemodelsofinlandwatersystemssimulatetheflowinriverbranchesbysolvingthe one‐dimensional or quasi‐two dimensional shallow water equations, also known asSaint‐Vernantequations(Deltares2012a,b).Thefact thatthesehydrostaticmodelsdonotsimulate the flowaroundhydraulicstructuresexplicitly isnotasevere limitation formostapplications. The system effect of the operation of various gates on the water levels inadjacent water bodies (river branches) can be studied, for instance (Becker andSchwanenberg, 2012). For stability of granular bed material and salt water transport,however, the flowacceleration in theverticaldimensionneeds tobesimulated.Moreover,the downside of primarilywater level‐centered validation and calibration in combinationwith parameterized structure representations (such as constant discharge coefficients) isthatthepredictionqualityofdischargesinsystem‐scalemodelsisoftenunclear.Warminketal. (2007,2008) investigated theuncertainty in calibrationofwater levels in rivermodelsresultingfromthelimitedavailabilityofdischargedata.Itwasconcludedthatthenecessaryextrapolation of the calibration parameter (bed roughness of main channel) leads tosignificant uncertainty in simulated design water levels. More intensive measurement ofdischarges, for which most gated structures are ideal, and a physically more realisticrepresentation of hydraulic structures in models are self‐evident improvements thatneverthelessrequireacultureshift.Theremainderofthischapterisorganizedasfollows:first,wedescribetheoverallapproach,thenthemethodisdescribedinthreesectionsaboutdischargemodelling,free‐surfaceflowsimulation and analysis of themodelling results.Next, the results of a series of validationruns for the free‐surfacemodel arediscussed, followedby resultsof a test case that givesnumerical examples of all modelling steps. We end the chapter with recommendations,conclusionsandanoutlookonfuturework.
4.2 Approachandmethod
4.2.1 GeneralForobtainingatimelypredictionoftheflowaroundgates,weproposeamulti‐stepphysics‐basedmodellingstrategywhichusesdatainputfromasystem‐scalemodel.Thework‐flowofthesuggestedgateoperationsystemisshowninFigure4.1.
57
Figure 4.1. Scheme of evaluation steps leading to a decision on optimal gate operation. Steps 1‐4 aretreatedinthischapter.Thedashedlineshowstheshorterdecisionsequencetakenbybarriersystemsthatdonottakeintoaccountfloweffects.
The first step consists of the extraction of predicted water levels on both sides of thestructurefromafar‐field(system‐scale)modelthatcontainsthestructure.Differentpossiblegate settings (when toopen,howmanygates touse)are identified in the secondstep.Alloptions need to be assessed in termsof discharge capacity; this happens in step 3. In thefourth step of Figure 4.1, for all gate configurations capable of discharging the requiredvolume, the resulting flow is simulated using CFD. Subsequent analysis of the simulationresultsdetermines the impactof the flowforspecific issuessuchasbedstability.The fifthand final step comprises the actual decision of gate operation actions. The conventionalsequenceofstepstakenbymostoperationalsystemsfollowsthedashedlineinFigure4.1,skipping steps 3 and 4. The present study focuses on steps 2‐4,which can be seen as anadditiontocomputationaldecisionsupportsystems(steps1and5)byBoukhanovskyandIvanov(2012)andIvanovetal.(2012).Amulti‐gateddischargesluicewithunderflowgateswillbeusedtodescribethemodellingmethod.Thecentralquestionaddressedishowtofindthesetofgateconfigurationscapableof delivering the required discharge that also meet the relevant constraints on flowproperties.4.2.2 Configurationsofmulti‐gatedstructureLet us consider the gate configurations of a discharge structure consisting of n similaropenings, each accommodating amovable gate. See Figure4.2. In its idle state, alln gatescloseofftheopeningsbetweenthepiersandthetotaldischargeiszero.Duringadischargeevent,mgateswillbeopenedpartiallyorcompletely,allowingacertaindischargethroughthestructure.A'gateconfiguration'isdefinedastheallocationofanumberofgates(m≤n)that are opened with a gate opening a(t) while the other gates remain closed. All gatesselectedforopeningwillbeoperatedsimilarly,i.e.withthesamea(t).
1. predict future water
levels upstream and downstream
3. determine gate settings
that meet required
discharge volume
4. simulate and analyse gate
flow and determine
optimal gate configuration.
2. determine set of possible
gate configurations
ledom egrahcsidledom metsysfree-surface model
& flow analysis
5. decision support system
for gate operation
58
Figure4.2.Amulti‐gateddischargesluiceinplanview.Inthisexample,gates3,4,5areopened,theothersareclosed;son=7andm=3.Thedottedlinedepictsplaneofsymmetry.
Before decidingwhich gates to open, first the possible combinations of opening gates areidentified and counted. In general, flow instabilities arenot favourable formaintaining anefficient and controllable discharge. As in other parts of physics, symmetry is a globalmeasureforstabilityof free‐surface flows. Ifasymmetry isallowed,mgatescanbechosenfreely from the total of n available slots. Then the number of possible combinations isobviously ,usingthecommonnotationforcombinatorialchoiceofmobjectsoutofn.Fortheconditionofsymmetrytohold,gatesmayonlybeopenedinsuchawaythatthepatternis symmetric about the vertical planeof symmetry in flowdirection (see Figure4.2). This
impliesthatthenumberofoptionsreducesto ⁄⁄ forall0 ,wheremcannotbe
chosenoddifniseven–inwhichcasetherearenooptionsatall.Forastructurewithsevengates(n=7), for instance,thetotalnumberofpossiblewaysto
open1,2,..,7gatesis∑ 1 2 1 127ifasymmetryisallowedand∑ ⁄⁄
2 ⁄ 1 15ifonlysymmetricconfigurationsarepermitted.Thisshowsthatthesymmetryconstraintgreatlyreducesthenumberofwaystoopenagivennumber of gates. Furthermore, an even number of gates has roughly half the number ofpossibilities, becauseopeninganyoddnumberof gates then results inasymmetric inflow.Thiscouldalsoholdforanodd‐numberedgatestructurewhichmissesone(oranyoddm<n)ofthegatesduetomaintenanceoroperationalfailure.4.2.3 SystemmodelandgatecontrolThebasisisformedbyaclassicboxmodel,seee.g.StellingandBooij(1999).Thefocusisonsubmergedflowthroughamulti‐gatedoutletbarrierthatblocksseawaterfromenteringthelake at high tide and discharges riverwater to sea at low tide, see Figure 4.3. This basicmodel serves in the present study as a surrogate system‐scalemodel. Thewater levels itgenerateswillbeusedasboundaryconditionsforthenear‐fieldmodelling.
GATE 1
lake
sea
w
pier
GATE 2 GATE 6 GATE 7GATE 3 GATE 4 GATE 5
59
Figure4.3.Classicboxmodelofoutflowofariver tosea.Anoutletbarrierstructureregulates the lakelevelwhilekeepingsaltseawaterout.Assumingbarriergatesareclosed(exceptwhendischargingundernaturalheadfromlaketosea)andassumingzeroevaporationandprecipitation,thesystemisdescribedby:
, (4.1)
whereQriver isdischarge fromariver,Qbarrier is the totaldischargethroughthegatesof thebarrier,hlakeisthewaterlevelinthelake,Alakeistheareaofthelakeassumedindependentofhlake. Submerged flow past an underflow gate is by definition affected by the downstreamwaterlevel.AsSection4.2described,theassociateddischargedependsonbothwaterlevels(sea and lake), the gate openinga and a discharge coefficient for submerged flowCD thatincludes vertical contraction effect. The discharge Q through a barrier gate at time t iswrittenas
2 , (4.2)
wherewistheflowwidth(seeFigure4.2)andthesubscript“barrier”isdroppedfromnowon.Sealevelhseaisapproximatedbyasinefunction.Thetotaldischargedvolumethatpassesthebarrierintheperiodduringwhichhlake>hseaisfoundaftersummingoverallmgatesandintegratingwithrespecttotime.Twogateopeningscenarioswillbeconsidered. Inbothscenariosequalgateopeningsa(t)are applied to allm gates selected for opening. The first scenario uses a constant gateopeningaconst for thewhole discharge period (from tstart to tend). The opening required tolowerthelakeleveltoadesiredlakelevelhtargetisfoundbyestimatingtheaveragerequireddischargeQtot,reqtoachievethisandbymakingestimatesoftheaveragedischargecoefficientandwaterlevelsduringthedischargeperiod:
sea
Qriverlake
hlake Alake
hsea
60
,
2with
,,
, (4.3)
wherebarsaretime‐averagesandprimesindicatepredictionsoffuturevalues.Inthesecondscenario, the discharge is regulated by a proportional integral derivative (PID) controller(Brown,2007).Thegoalofthisscenarioistohaveamoreconstantgatedischargebyvaryingthegateopenings intime,whilststillachievingthesamehtargetas inthe firstscenario.ThediscretePIDformulafordischargeattiis
∆
, (4.4)
whereKiarethegainparametersandtheerrorvalueisdefinedas .Inthesimulations,KP=0.10,KI=0.45andKD=0.55areused.ThesetpointQsetisconstantandequaltoQtot,req,exceptforlinearsetpointrampingappliedatthestartofdischargetopreventunduefluctuationsofgateposition.Ateachtimestep,therequiredgateopeningisderivedfrom thisdischargedividedby 2 . Figure4.4 captures the flowchartofthesystemmodel.Itincludescomputationsofthetwogateoperationscenarios.
Figure4.4.Systemmodel:flowchartofgatecontrolandwaterlevelcomputations.Figure4.4showsthatthetotaldischargecomputedbythesystemmodelQtotisbeingusedtocalculatethenewlakelevel.Additionally,itshowsthatatthestartofeachdischargeevent,
determine hsea and
Qriver
hsea < hlake?
discharge model
calculatelocal water
levels, achieved gate discharge
and CD
YES
mhlake(t0)htargetAlake
NO
Qgate(t), Qtot(t)a(t)
hlake(t), hsea(t)
calculateQtot = m Qgate
Qnet = Qriver – Qgate
calculate hlake
all gates closed, Qgate = 0
time loop
input
output
determinegate opening a(t) that gives desired
dischargeapply
discharge control?
PIDdischarge control YES
NOcalculate
t’endQ’tot,req
update CD’
start of discharge
event?
YES
NO
*
61
i.e.whenthegatesareopened,thepredictionofthedischargecoefficientCD´isupdatedusingdata from the discharge model. For both situations, with and without PID control, thiscoefficient is foundby a relaxation formulawith themeandischarge coefficientCD* of thepreviousdischargeeventcomputedbythedischargemodel.Forthen‐thdischargeeventtheupdateformulareads
1 ∗ 1 1 (4.5)
In all computations a relaxation factor β = 0.75 is applied. Discharge coefficients actuallydependonnumerous factors.Also, flows throughneighbouringgates influenceeachother.Todistinguishbetweendifferentgateconfigurationswiththesametotalflow‐througharea∙ ∙ , thesetwothingsneedtobetaken intoaccount.This isdone inthedischarge
modeldescribedinthenextsubsection,seealsotheboldblockinFigure4.4.4.2.4 DischargemodelVerticalliftgateswithunderflowareraisedverticallybetweenpiersofastructure.Whenthegatesare liftedhigher than thewatersurface, thereexists freeorsubmergedVenturi flow(Boiten, 1994). These flow types have different discharge characteristics and associatedformulae.Forestimatingthesubmergedflowdischargewithapartlyloweredgate,thelocalwaterdepthsareschematizedaccordingtoFigure4.5(afterKolkman,1994).
Figure4.5.Definitionsof localwaterdepthshi forunderflowgate inhydraulicstructure,afterKolkman(1994).Above:topviewofpier;below:crosssectionfreewatersurfacearoundgate.Sketchnottoscale.
Bernoulli’s equationand themomentumbalance togethergivea systemof fourequations.Eachequationisdenoted[hi,hi+1]anddescribesthetransitionfromwaterlevelhitohi+1(hi+1beinglocateddownstreamofhi),inlinewithFigure4.5.
a
gate body
h0h3
connection to barrier structure
main flow direction
Сс a
h2
h1 h4
pier between gates (top view)
sill height
sea level
lake level
z’ = 0
z = 0
62
, : 2 1 2
, : 2 2
, :12
12
h , h :2
1 2
where1
,1 with , 0.025 1 0.6 (4.6)
Theseequationsaresolvedfromtheirpolynomialforms,whichworksbecausethesmallestpositive root gives the desired water level in each case. In order of computation in thedischargemodel,theseformsare:
2 1 2 0
, 2 2
2
1 2 0
12 ,
12
0. (4.7)
These polynomials are solved for 0 <Q ≤QMF. The constraint h2,forward =h2,backwardmay besolvedgraphically,yieldingtheactuallyachieveddischarge.Transitionsh0–h1andh3–h4withlosscoefficientsξinandξoutrepresenttheeffectsofflowenteringandleavingthenarrowareabetween two piers. Transitions h1–h2–h3 are the characteristic underflow gate zones.Computationswerecarriedoutaccording to the flowchart inFigure4.6.The lakeandsealevelscomputedinthesystemmodelservedasboundaryconditions–forvariablesh0andh4ofthismodel,respectively.
63
Figure 4.6. Flow chart of discharge model. This computation is repeated each time step; it is fullycontainedintheblocknamed“dischargemodel”inFigure4.4.
Agoodgeometricdesignofadischarge‐regulatorissuchthatnotransitionoccursfromoneflowtypetoanotherduringregularusage.Themodelthereforechecksifindeedsubmergeddischarge occurs. As criterion for reaching themodular flow dischargeQMF, theminimumflowdepthinthecontrolsectionh2iscomparedtotheflowheightinthepointofmaximumverticalcontractionCc∙a.Freeandintermediateflowregimesarethusdetected,butarenotbeing calculated. Submerged Venturi flow is not considered either, since the idea is toactivelycontroltheflow.All four non‐linear equations are reshaped into third‐order polynomials f(hi, hi+1, Q) = 0.DischargeQ is substituted for the velocity termsand remains as theonlyunknown in thesystem of equations. As prescribed for sub‐critical flow conditions (Chow, 1959),computationaldirectionbehindthegateisfromdownstreamtoupstream(h4toh2).Onthelakeside,computationsgoinflowdirectionuptothecontrolsection(h0toh2).ThedischargecoefficientCD isderived fromthecontractioncoefficientCc forsharp‐edgedgates, fittedonexperimental data cited inKolkman (1994) so that the full range of gate openingsa/h1 iscovered.Combiningtwoempiricalformulations,thecontractioncoefficientforsharp‐edgedgatesisassumedequalto:
0.782
1.782 ⁄for 0.5 regime Henry, 1950 and
0.004 log ⁄ 0.6074 for 0.5 regime Cozzo, 1978 . (4.8)
IterationsonQultimatelyyieldavalueatwhichh2,forward,computedfromupstream,isequaltoh2,backwardcomputedfromdownstream.This is theachievedvalueofQ for thegivengateopeninga.Theentranceandexitlossesareassumedtodependonthenumberofgatesinuse(m).Themethoddoesnotdistinguishbetweendifferentgate configurationswithequalm,however.Numericalresultsareshownintheresultssection.
determine modular flow
limith2 ≤ Cc*a
?QMF
found
Q < QMF exists at
which h2's match?
YES YESm
hlakehsea
a
NO
QDMCD
iterate on discharge Q
NO
h0→h1h1→h2,forward
h4→h3h3→h2,backward
compare h2's
system of gate equations
from Q = 0 to QMF
Q found for
current a, hlake,
hsea
no submerged flow at this gate opening
inputoutput
until h2,forward = h2,backward
*
determine ξin, ξout
64
4.3 CFDsimulations
Step4 inFigure4.1consistsof twoparts: free‐surfaceCFDsimulations,called“Model I” inTable3.1anddiscussedinSection4.4.1;andflowimpactanalysis,discussedinSection4.4.2.4.3.1 Modelset‐upA non‐hydrostatic flowmodel is applied to find outwhich of the selected gate settings ismost favourable in termsof flowproperties. The two‐dimensionaldomain is definedby averticalcrosssectionthroughthegatesection(abbreviatedas2DV)fromalaketothesea,seeFigure4.8.Arigidrectangulargatewithasharp‐edgedbottomismodelledimplicitlybycutting its shape out of the flow domain. The Reynolds‐Averaged Navier‐Stokes (RANS)equations for incompressible floware thebasis for thesimulations.Thesewillbegiven inChapter6.Figure4.7givestheflowchartoftheCFDsimulations.Themodeldomaincoverstheflowfromh1toh3.Theseinputvaluesaretakenfromthedischargemodel.
Figure4.7.FlowchartofFEMfree‐surfaceflowsimulations.
Foreachsimulatedflowsituationtwoconsecutiverunsaremade:asteady‐staterunandatime‐dependent transient run. In the former run, iterationson theoutflowvelocityprofileare done until pressure at the surface becomes zero. The results of this pre‐run are thenimplemented as initial conditions for the transient run, which uses a moving mesh tosimulate the free surface. Boundary conditions are similar for both runs except for thesurfacedownstreamofthegate,seeFigure4.8.
Figure4.8.BoundaryconditionsofCFDmodel.Themainflowdirectionisfromlefttoright.Notdrawntoscale.The upstream flowboundary consists of a hydrostatic pressure profile .Thedownstreamboundaryisablockprofileu‐velocity.Noslipisappliedatthewalls(u 0)alongwithawallfunction.Thesteadypre‐runusesa‘rigidlid’(freeslipboundary, ∙ 0)
steady-state run with rigid lid
pressure = 0 at surface at
outlet?
transient run with ALE-mesh for free surface
convergence of h3, Q and TKE ?
YES YESh1h3a
Qgate
NO start CFDanalysis
if p > 0 then increase u(z)if p < 0 then decrease u(z)
simulate longer
NO
input
initial guess of u(z)
output
u, p, k, εfree surface
free slip in steady-state run;open boundary in transient run
no slip &wall function
u-velocity profileno slip &
wall functionhydrostatic pressure
main flow direction
free slip in steady-state runand in transient run
GA
TE
xz
0
65
for thedownstreamwater surface.Theupstream free surface ismodelled as a rigid lid inbothruns.An unstructured computationalmesh is usedwith refinements near the bottomwall andgateboundaries,madeupofaround35,000triangularelementsandyieldingabout230,000degreesof freedomforatransientrun.TheArbitraryLangrangian‐Eulerian(ALE)method,see previous chapter, is used to compute the deformation of the computational mesh.Winslowsmoothing (Doneaetal., 2004) is applied to find the locationof the innernodes.Figure 4.9 shows part of themesh. At the top boundary in the transient run, the velocitycondition isanopenboundarywithzerostress innormaldirection.At thesameboundarythemeshvelocityinnormaldirectionisprescribedas , ∙ ∙ (FerzigerandPerić,2002).Meshconvergencetestsshowedthattheappliedmeshissufficientlydensesothatresultsdonotimproveonfurthermeshrefinement.
Figure4.9.Examplesnapshotshowingpartofthecomputationalmesh.Adeformedfreesurfaceboundarydownstreamofthegateisvisible.Flowisfromlefttoright,unitsinmeters.The more common choice of applying a velocity condition upstream and a pressureboundarydownstreamconflictswiththerequiredALEmovingmeshconditionattheoutletboundary. Vertical mesh freedom is necessary for the surface movement. A hydrostaticpressureprofilecannotbeprescribedattheoutlet,sinceanychangeinwaterdepthatthisboundarywould imply a change of local pressure,which contradicts the applied pressureprofile.Inthecourseofthetransientrun,thefreesurfaceadaptstothepressurefieldandviceversa.Becausethephysicalflowsituationisquasi‐steady,withfluctuationsdependingondegreeofsubmergenceandgateopening,thesurfacemayshowoscillationsintimeinitsequilibriumstate. As a consequence the flowdischarge is also not strictly constant in the equilibriumstate.ThepackageCOMSOLMultiphysics(version4.2)isusedtosimulatethegateflow(COMSOL,2013). This finite element method (FEM) solver is applied to solve the discretised RANSequations.Thegeneralizedalphatime‐implicitsteppingmethodisappliedtoensureCourantstability,withastrictmaximumtimestepof∆t=0.02s.Thetimestep intheCFDmodel iscompletely independent of the time step in the system model and discharge model. The
66
variablesare solved in two segregatedgroupsusing a combinationof thePARDISO solverand the iterative BiCGStab solver in combination with a VANKA preconditioner. Thestandardk‐εmodelisusedforturbulenceclosure.Simulationof24secondsofphysicaltimetookaroundsixhoursofwall‐clocktimeonIntel8‐corei7processor,2.93GHz,8GbRAM,occupyingonaverage1GbRAMand50%oftotalCPUpower.4.3.2 Analysisofsimulationoutput:flowimpactThesecondpartofstep4inFigure4.1isthepostanalysisofthemodellingresultsobtainedinprevioussteps.Threeaspectsarediscussed:flowparameters,vibrationsandbedstability.FlowparametersThreeparametersthatarerequiredforassessingvarioustypesofflowimpactareextractedfromtheCFDmodel:thecontractioncoefficientCc,thevelocityinthevenacontractaUvcandtheFroudenumberFr.Theflowfieldisinterpolatedtoaregulargrid,sothattheedgeoftheseparatedlayerisfound,seeFigure4.13.Thecontractioncoefficientisthusfounddirectly.Thecross‐sectionalaveragedvelocity in thevenacontractadefinedbyaspatialaverage intheseparatedshearzone,isafunctionoftimehere:
1
,∙
, (4.9)
whereUisthevelocitymagnitudescalaratthepointofmaximumflowcontraction.Forgateflowwithsignificantfluctuations,thetemporalmeanofthisquantity, ,maybeused.TheFroudenumberFrisusedasanaidforfindingthetransitionstointermediateandfreeflowregimes.Itisconsideredtime‐dependenttoo:
Fr , (4.10)
inwhichobviously ∙ in fully free flow.Anoverviewofcritical flowtheory fromahistoricalperspectiveisgivenbyCastro‐OrgazandHager(2010)andfromamorepracticalviewpoint by Boiten (1994). In a more complete flow assessment, not only the verticalcontractioncausedbytheunderflowgateisusedasacriterionformodularflow,asisdonehere,butalsocontractioncausedbyhorizontalandpossiblyverticalflowdomaintransitionsattheinletofthestructureshouldbeincluded.VibrationsAs a simple estimating factor for occurrence of FIVs, a time‐dependent version of thereducedflowvelocityVr(Section2.4)isintroduced:
∙
, (4.11)
wherefgateistheresponsefrequencyofthestructureinHz;Disthegatethickness.Thebestoptiontofindthegatefrequencyistomeasureitinsitubyinstallingsensors.Anelaboration
67
of this idea is discussed in Chapter 7. In a first, crude estimate of the risk of a vibrationinitiating, it canbeassumed that theamplitudeA due toFIV is a functionofVr,a and thelowerwaterlevel: , , .ScourandbedprotectionTheclassicalpredictionoflocalscourdownstreamofweirsandsluicestructurescausedbyoutlet currents isdescribedbyBreusers (1966)andHoffmansandPilarczyk (1995).Morerecently,contemporarycomputationaltechniqueswereintroducedforscourestimation,e.g.Azmathullah et al. (2006). In the classical physics‐based design formulae, turbulenceparameters are used topredict thedepth of the scourhole in unprotectedbeds. For bedsprotected with granular material (loose rocks), the Shields parameter is a classic non‐dimensionalmeasureappliedasa first indicator for instability (Shields1936).Anadaptedversionof thisparameterusedby Jongeling et al. (2003) andelaborateduponbyHofland(2005)andHoanetal.(2011)isdefinedas
Ψ⟨ α ⟩
Δwith Δ . (4.12)
where⟨. . ⟩denotesspatialaveragingoverthewholewaterdepth,k istheturbulentkineticenergy(TKE),disthelocalwaterdepth, isthemeanflowvelocitymagnitudeandαisanempiricalparameterforbringingintoaccounttheturbulence(thatdependsonflowtypeandlocalgeometry,e.g.slopesinbottomprofile).
4.4 Modelvalidation
Aseriesofvalidationrunswasdoneforthefree‐surfacemodel.‘Validationrun’isusedhereinthemeaningdiscussedbyStellingandBooij(1999):theuncalibratedmodelisrunwithoutanytweakingofparameterstoseeifitcanreproducethemostimportantphysicalfeatures.Experimental laboratorydatabyNago (1978,1983) for a vertical sharp‐edgedgateundersubmergedeffluxserveascomparison.Nago’s(1978,1983)dimensionswereusedwithoutanyscaling.Hisdischargeformula 2 doesnotcontainthedownstreamlevelh3explicitly. Its influence is instead incorporated inCE.The twocoefficientsareconnectedthrough
2
, (4.13)
The simulated discharge is computed by spatial integration of horizontal velocity at theoutflowboundary.InFigure4.10coefficientCEisplottedfordifferentseriesofdimensionlessgateopeningsandforarangeofdimensionlessdownstreamlevels.
68
Figure4.10.ResultsofvalidationrunsshowingdischargecoefficientCEsimulatedbythefree‐surfaceCFDmodel versus experimental data of submerged flow of a sharp‐edged underflow gate byNago (1978,1983).Above:Sortedbygateopening (a/h1)anddownstream level (h3/a).Below:directcomparisonofthesamedata.Dashedlinesmark10%deviation.Theresultsofthevalidationrunsmakeclearthatthesimulationscapturethedischargesoftheexperimentaldataquiteaccurately:thecorrelationcoefficientis0.994andtherootmeansquareerroris1.14%.Thefactthattheuncalibratedmodelshowsgooddischargeestimatesgives confidence in the predictive power of this modelling approach. Physical output notvalidatedhere(suchasTKE)maybecalibratedinfuturestudiesbyadjustingsuitablemodelparameters.Convergenceofvariousflowvariablesoccursatdifferentrates.First,themean
69
velocitiesstabilize,andthentheforcesonthegateconverge,thenthedischarge,andlastlytheturbulentenergy.Thechosenboundaryconditionsprovedtoleadtostableresultsforallsubmergenceratiosof Nago’s (1978, 1983) data. It was found that themovingmesh is the critical factor fornumericalstability.ALEisasuitablemethodforcomputingthefreesurfaceforquasi‐steadygate flowas long as the flow remains submerged. Steep surface gradients associatedwithloweringh3causeinvertedmeshelementsandhencenumericalinstabilities.
4.5 Resultsoftestcasesimulations
Thedescribedmethodsareillustratedbyatestcaseexample.Theresultsofthreemodellingstepsarediscussed:thesluicemodelcontainingthesystemmodel(forwaterlevels)plusthedischargemodel (Figures4.4 and4.6), the free‐surfacemodel (Figure4.7) andanalysis ofvibrationsandbedstability.Fourtidalcyclesandfourdischargeeventsweremodelledforadischargesluicewithsevengatesregulatingalakewithconstantriverinflow.Thegoalistodeterminetheoptimalnumberofgatestoopenandthebestgateoperationscenario.4.5.1 ResultsofsystemanddischargemodelModelparameters‐n=7,m=1,...,7‐Alake=1.9∙107m2‐Qriver=100m3/s‐hlake(t=0)=6.1m‐htarget=6.0m‐w=22.5m‐sillheight:3m‐meansealevel=z’+6.1m‐tidalamplitude=0.60m‐tidalperiod=12.5hours
70
Figure4.11.Resultsofsluicemodelfor3≤m≤7:seaandlakelevelforgateoperationscenariowithandwithoutPID‐controlleddischarge.Verticallineindicatesmomentofmaximumheaddifference.
Thesluicemodelwasrunfor1≤m≤7.Whenopeningonlyonegate, thetarget lake levelcouldnotbereachedevenwhenliftingthegatecompletely.Whenusingtwogates,thetargetlevelisreached,butthemodularflowlimitisexceededforthegreatestpartofthedischargeperiod.Thisresults inunwanted transitions to intermediateand free flowwith fluctuatingdischarges that are hard to control. For 3 ≤m≤ 7 strictly submerged flow exists and thetarget ismet. Therefore, only these configurations aremodelled further. Thewater levels(plottedforonecycleinFigure4.11)showthatthelakelevelfluctuatesinacontrolledwayand nearly identically for the scenarioswith andwithout discharge control.Note that thetargetlakelevelindicatesnotadesiredaveragelevel,butthelevelwherewewantthelaketobeattheendofadischargeevent.InthecaseofFigure4.11itisreachedexactlyaroundt=24.5h,andbothdischargemethodsareslightlyabovethetargetattheendofthenexteventaroundt=37.0h.
71
Figure4.12.Resultsofsluicemodelforusingthreegatesandsevengates:gateopenings(upperplot)andachieveddischargespergate(lowerplot).
InFigure4.12, thegateopeningsandachievedgatedischarges in timeareplotted foronetidalperiodforthesituationswiththreeorsevengatesopenedduringthedischargeevent.Intermediatenumbersofoperatedgates(4≤m≤6)liebetweentheshowncurvesform=3andm=7, but are not plotted for clarity. It can be seen that constant gate openings givedischargesthatvaryintimefollowingthetime‐dependenthydraulicheaddifference.Inthe
30 32 34 36 38 400
0.5
1
1.5
2
2.5
3
t (hours)
gate
ope
ning
a (
m)
3 gates
7 gates
run I
run II
run III
constant openingopening varied by PID control
72
PID‐controlledscenario, thegateopening isautomaticallyoperated in suchaway that thedischargestabilizesquicklyafterthestart.Inthismulti‐scalemodellingapproach,averagedvaluesfromthedischargemodelareusedto improve discharge predictions at system scale. However, instantaneous discharges andgateopeningscomputedinbothmodels inevitablydiffer.Largestdiscrepanciesarearound10%.Thiscouldbe improvedbyexaminingdifferentupdatemethods,atthecostof longercomputationtime.Threeconfigurationsareselectedforevaluationbyfree‐surfacesimulations.ThesecasesaremarkedinFigure4.12asrunsI,IIandIII.RunsIandIIIrepresentextremes:aconstantgateopeningwithonlythreegatesinuse(highQ)andacontrolledopeningwithallsevengatesinuse(lowQ).Allthreerunsareatthetimeofmaximumheaddifference.Inreal‐lifepractice,more cases could be selected for simulation depending on specific interests and availablecomputingpower.4.5.2 ResultsofCFDsimulationsTosimulatethetwoselectedrunsIandIIwithinthevalidatedrange,thelevelsandopeningarescaleddownwithlengthscale1:10,seeTable4.1.Table4.1.ValuesofselectedCFDruns.
run
gate
configuration
length
scale h0(m) h1(m) h3(m) h4(m)
gate
openinga
(m)
Im=3,
constantopening
1:1 3.07 2.93 2.38 2.50 1.30
1:10 0.307 0.293 0.238 0.250 0.130
IIm=3,
PIDcontrol
1:1 3.07 3.00 2.44 2.50 1.14
1:10 0.307 0.300 0.244 0.250 0.114
IIIm=7,
PIDcontrol
1:1 3.07 3.06 2.49 2.50 0.610
1:10 0.307 0.306 0.249 0.250 0.0610
inputvaluesforCFDruns.
Allwaterlevelshiarerelativetoz=0.
73
Table4.1(Continued).
run
totaldischargeQtot
(m3/s)
dischargepergate
Qi(m3/s)
dischargepergate,
perunitwidthqi(m2/s)
I270 90.1 4.00
0.855 0.285 0.127
II237 79.02 3.51
0.750 0.250 0.111
III237 33.86 1.51
0.750 0.107 0.0476
The near‐gate flow velocities, pressures, TKE and dissipation are simulated. Figure 4.13showsaplotofthesimulatedflowfieldofrunII(atlengthscale1:10)byindicating .
Figure4.13. Vector flow fieldofrunII.Flow is from lefttoright.Thecomputed freesurfacebehindthegate shows local lowering.Dashed line indicates separationbetweenpositiveandnegativeu1‐velocity.Thefigureshowsonlypartoftheactualcomputationaldomain.Totaldomainlengthis3.6m.Thesimulatedfreesurfaceasexpectedsinksintheregiondirectlydownstreamofthegate(solid line in Figure 4.13). In this case, the vena contracta is located at short distancedownstream of the flow separation point. The separation between positive and negativehorizontal velocities in the recirculation area is derived (dashed line in Figure 4.13). At adistanceofaroundfivetimesthedownstreamwaterlevelpastthegate,theflowreattachesatthesurfaceandthevelocitystartstoreturntoamoreuniformprofile.Figure4.14showsplotsofthepressureandturbulentkineticenergyofrunII.
74
Figure4.14.PressurepinPa(above)andturbulentkineticenergyk(TKE)inm2/s2(below)ofrunII.In the case shown in the plots, the equilibrium state reached in the simulations is fullysteady.Pressuregradientsaremild;thepressurereturnssmoothlytoahydrostaticshapeasthestreamlinesbecomeparalleldownstream.TheTKEreachesamaximuminthemiddleofthewatercolumnatabouttwotimesthedownstreamwaterdepthpastthegate.RunIhasasteepersurfacebehindthegatethanrunII(showninFigures4.13and4.14)andhigherTKElevels,whilerunIIIhasthelowestTKElevelsandthemostlevelsurfacedownstreamofthegate.4.5.3 ResultsofflowanalysisTheoutputoftheCFDfree‐surfacemodelisusedforcomputingthevaluesofthethreeflowparametersthatwerediscussedinanearliersection,seeTable4.2.Table4.2.ComputedflowparametersderivedfromCFDmodelresults.
run Cc(‐) Uvc(m/s) Fr(‐)
I 0.88 3.56 0.83
II 0.86 3.50 0.78
III 0.84 2.74 0.57Table4.2showsthat thecontractioncoefficientsdonotdiffermuch,which isexpected forsimilar gate types. The velocity in the control sectionUvc is highest for the situationwithhighestdischargepergate(run I)and lowest for thesituationwithsmallestdischargepergate(runIII).Thesameholds fortheFroudenumber.Thismatchesobservations fromthefreesurfacecurvaturesofthefinalsolutionofthetransientsimulations.The flow impact on the bed protection material is estimated by computing Ψ for twodifferent α for the selected runs, see Equation (4.12). Variation of α controls the relativecontributionofturbulentkineticenergytotheflowimpact.Thewholewaterdepthdisused
for averaging the square of themaximum local velocity term √ . The results areplottedinFigure4.15.
75
Figure 4.15. Computed values of bed stability parameter Ψ downstream of the gate for two differentvaluesofturbulenceimpactparameterα.RunsI,IIandIIIareshown.TheplotshowsthatrunI(threegateswithconstantopening)hasthestrongestflowimpactonthebedmaterialofthethreerunsirrespectiveofthechoiceforα.TheΨ–valuesofrunIIshow that controlling thedischargewithout openingmore gates alreadyhas a lower flowimpact on the bed. Run III (seven gates with controlled discharge) has the lowest flowimpact. All runs reach theirmaximum flow impact on the bed around the same (limited)distancedownstreamofthegate.Forallrunsthegeneralshapeofthecurvesisquitesimilarforbothvaluesofα, indicatingthatturbulenceisdominantovermeanvelocityfortheflowimpact.Overall thevaluesof thebedstabilityparameterare somehwat lowcompared topreviousnumerical investigations by Erdbrink and Jongeling (2008) and Erdbrink (2009), whichcouldbeattributedtotheuseofthestandardk‐εmodelinthisstudyinsteadoftheRNGk‐εturbulence model used in the two mentioned studies. Choosing higher α values couldcompensate the lower TKE. For practical application one should fix α after calibration inexperimental investigations and one should define a threshold value for Ψ not to beexceededduringoperationtobeusedformeasuringthefitnessofdifferentflowscenarios.Turning to the assessment of gate vibrations, a measured range of structural responsefrequencieswouldgivethereducedvelocitynumber.Ifresponsecharacteristicsofthegatewouldbeknown,thiswouldyieldanestimateofvibrationlevels.Fornow,thiscomputationispostponed;inthenextchaptersuchresponsedatawillbefoundexperimentally.Basedonthediscussedmodellingresultsandflowanalysis,itmaybedecidedtoimplementthedischargescenarioofrunII,becauseitgivesalowerimpactonthebedmaterialthanrunI–whilestillensuringsufficientdischargevolumetoreachthetargetlakelevel.
76
4.6 Modelcouplingtests
In the calculationofaconst at the start of discharge, somevalue forCD is needed. From thesecond discharge event onwards there is data available from the discharge model onpreviouslyachieveddischarges.Thisdatacomputesthedischargesmoreaccuratelythanthesystemmodel, because it solves the local flow equations (that include contraction, losses,etc.)andhenceworkswithonevalueofCDforeachtimestep,ratherthankeepingitconstantthoughoutthedischargeeventasdoesthesystemmodel.SixscenariosweretestedforusinginformationfromthedischargeinthesystemmodelforestimatingCD. For each scenario the lake level is updatedusing thedischarge through thestructureascomputedbythesystemmodel.Alltestrunsusefivegates(m=5).Table4.3andTable4.4containthedefinitionsandtheachievedresultsofthetestedcouplingscenarios.Table4.3.Testingscenariosformodelcoupling.
scenario
PIDdischargecontrol
targetlevel
reached?
dischargemodel
resultsusedinsystemmodel?
convergenceofCD
convergenceofa
convergenceofQ
1 off yes no no no no2 off yes yes yes* no yes3 off yes yes no yes no4 off yes yes yes maybe yes5 on yes yes yes* yes no6 on yes yes yes yes yes
*Averagevaluesconverge.Table4.4.Resultsoftestingscenarios.
scenario
openinginsystem
modelaSM(m)
openingindischargemodelaDM
(m)
error|aSM–aDM|⁄aDM(%)
discharge insystemmodel QSM(m3/s)
dischargeindischargemodelQDM(m3/s)
error|QSM–QDM|⁄QDM(%)
1 0.80 1.04 23 270 230 172 0.93 1.04 11 270 279 33 0.89 0.87 2 284 252 134 0.92 1.03 11 269 275 25 0.73 0.83 12 229 208 106 0.81 0.77 5 231 231.5 0
A good couplingmethod feeds the systemmodelwith enough information to improve itsestimatesofQgate,aandCD.CouplingerrorsareexpressedinTable4.4asrelativedifferencesatthetimeofmaximumheaddifferenceinthefourthdischargeevent.Whenconstantgateopenings are applied, the tests show that without updating the discharge coefficient, thesystem model is unable to make accurate estimates of the required gate opening anddischarges.
77
Testingall scenariosrevealed thatscenario6gives thebest results in termsoferrors.SeeFigure 4.16. This alternative offers the bestway to control the discharge and at the sametime obtain the best possiblematch between themodels. But the price for this is amorecomplextrimmingoftheupdateparameters(threePIDgainparametersandα)andlongercomputationtime.
Figure4.16.Testresultsofscenario6.Dottedlineindicatesdischargemodelandcontinuouslineindicatessystemmodel.Forthechoicewhichscenariosareappliedinthemainstudytheeaseofparametertrimmingandthestabilityofmaximumdischargemaximawereconsideredmostimportant.Thisledtochoosingscenario2fortheuncontrolledrunsandscenario5forthecontrolledruns.
78
4.7 Conclusionsofthischapter
Thedescribedcaseofamulti‐gatedoutletbarriersluicehasshownhowdischargeestimatesfromelementaryflowequationsandfree‐surfacetime‐dependentCFDsimulationscanaidindeciding on optimal gate configuration and opening scenarios. This introduces dischargecomputations inoperationalmanagementofhydraulicstructuresfor issuesthatatpresentaredecideduponbyjudgementoftheoperator.Themethodconnectsmodellingscaleswitha minimum of data exchange (coupling) and applies a PID‐controller to achieve a moreconstant discharge throughout the entire discharge event under natural head difference.Validationrunsshowedthatthefree‐surfacemodelproducesdischargevaluesforarangeofgateopeningsandsubmergencelevelswithinanacceptableaccuracyofexperimentalvalues.Applying PID‐control to discharge instead of to water levels is more complicated andrequires more gate adjustments (optimisation of this was not investigated here), but itachieves something that PID‐control on water levels cannot achieve: control of the flowimpactonthestructureanditssurroundings.
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5 Physicalexperiment3
5.1 Preface
Thischapterandthenextareconnectedbyacommonaim:toinvestigatethedynamicsofanewgatelay‐out.ThephysicalmodeldiscussedinthischapterwaspreparedandexecutedatDeltaresinDelft,TheNetherlands.Amoredetaileddescriptionoftheexperimentisreportedin Erdbrink (2012a). Chapter 6 continues the study of the same gate shape by numericalphysics‐based simulation of its FIV response. This will give an idea of the feasibility ofapplying fundamental numerical models as assessment tools for gate designs, as acomplementary toolnext tophysicalmodelling. In thepresent chapter, the set‐upand theresults of the physical scale model experiment are given. Conclusions and a morecomprehensivediscussionareincludedinChapter6.
5.2 Introduction
AslaidoutinChapter2,thedynamicresponseofahydraulicgateduetoitsinteractionwiththeflowstronglydependsondetailsofthegatebottomgeometry.Numerousexperimentalstudies of flow‐induced vibrations (FIV) of gates have previously looked into thecharacteristics of gate shapes (Hardwick 1974, Vrijer 1979, Kolkman 1984). The gainedinsightinexcitationmechanismshasresultedinwidespreadrulesofthumbforunfavourabledesigns that should be avoided as well as favourable design features (e.g. Thang 1990,Naudascher and Rockwell, 1994). However, fundamental knowledge and practicalexperience have not culminated in one ideal universal shape – partly because thesurroundingstructureisanimportantfactor.Experimentalandnumericalmodelsareincapableofcapturingalldegreesoffreedom(d.o.f.)experienced by real‐life gates (mass‐vibration mode in cross‐flow and in‐flow direction,bending, torsion). Streamwise (horizontal) vibrations are usually studied separately (e.g.Jongeling 1988) and sometimes in combination with the cross‐flow mode (Billeter andStaubli,2000).Inthisstudyweconsiderthemostfrequentlyencounteredandinvestigatedmodeforavertical‐liftgate:oned.o.f.inthecross‐flowdirection.Theemergenceandseverityof flow‐relateddynamic forcesonthegatearerelatedto flowinstabilitiesandbodymotioneffects(Section2.4).Fluctuationsoftheseparatedflow’sshearlayer may incite a mechanism called Impinging Leading Edge Vibrations (ILEV) for gateswithasharpupstreamedge.Ifthegatebottomhasanextendinglipinstreamwisedirection,the shear layer separated from the upstream edgemay reattach to the gate bottom in anunstableway, givingdynamic excitation. In a differentmechanism, periodic forces are theresult of initially small gate movements. This self‐exciting process is called Movement‐InducedExcitation(MIE).
3Thischapterisbasedonandusestextandcontentfrom“Reducingcross‐flowvibrationsofunderflowgates: experiments and numerical studies” by C.D. Erdbrink, V.V. Krzhizhanovskaya, P.M.A. Sloot,currentlyunderreviewattheJournalofFluidsandStructures.
80
Previous investigations have proved that most severe vibrations of underflow gates insubmerged flow occur at small gate openings and are predominantly caused by ILEV andMIEmechanisms(Hardwick,1974;ThangandNaudascher,1986aand1986b).Thecurrentinvestigation therefore focuses on small gate openings and does not look at the distinctlydifferentmechanismofnoiseexcitation.OthernotableexperimentalstudiesareKapurandReynolds(1967),NaudascherandRockwell(1980),Thang(1990),Kanneetal.(1991),Ishii(1992)andGomesetal.(2001).Onlythelaststudyalsoincludesnumericalmodelling.Assuming that adding structuraldampingoravoiding critical gateopeningsareunfeasibleoptions, the shape of the gate bottom is the decisive factor determining the tendency tovibrate.Iftheflowpassesthegatewhileremainingattached,orifthereisafixedseparationpoint and a stable reattachment, or if the shear layer is kept away from thebottom in allcircumstances,thentheILEVmechanismmaybeavoided.Athin,sharp‐edgedgeometrywithseparation from the trailing edge is favourable (e.g. the rightmost profile in Figure 2 inSection 2.2.2), because potential shear layer instabilities occur downstream from the gateand a small bottom area inhibits the occurrence of large lift forces on the gate, thusminimizing theriskofMIEvibrations.Butsuchadesign isoftennotpossibledue tootherdesignconstraints.Atthestartofthisstudyanumberofnewideasforattenuationmeaureswereidentified4:(i). Counter‐balance the vibrating gate mass by adding an extra elastically mounted
weighttothegatethatissetintomotiontocompensatethegatemovement.(ii).Tondl(1998)hasmadeseveralanalyticalstudiesintothequenchingofself‐excited
vibrations by varying the stiffness of the support (this is called parametricexcitation).
(iii).Influencethehydrodynamicgatepressuresactivelybyinjectingorsuckingwaterthroughapump‐regulatedsystemoftubesflowingoutthroughthegatebottom.Thisshouldworktodisturbtheexcitations.
(iv).Make holes (or shafts or slots) in the gate bottom profile so that an intentionalleakage flowcandevelop, againwith the goal of influencing thebottompressuresbeneficially.
(v).Adjustthebottomgeometryinsuchawaythattheprofileactsasa(semi‐)aerofoil.Thiswouldenableactive ‘steering’ofthegatebyusing(rotationsof)theprofiletoincrease or reduce the steady lift force. This acts as an aid for gate lifting andloweringunderaheaddifferenceandmayhelptoavoidcriticalgateopeningswherevibrationsoccur.
As far as known from literature studies, none of these measures have been investigatedbeforeor tested for theirachievability foruse inhydraulicstructures.The investigationathand chooses measure (iv) as the central new idea. An unfavourable thick flat‐bottomrectangulargateistakenasareferencegateandanewdesignwithleakageflowthroughtwoopeningsinthebottomsectionisinvestigatedasapotentialwayofimprovingthevibrationproperties,seeFigure5.1.
4 Ideas (i) and (iii)‐(v) originate from discussions at Deltares with Tom Jongeling, who helpedenormouslyinthedesignstageoftheexperiment.
81
Figure 5.1. Streamwise cross‐section of gate configuration showing ventilated gate design (detail ofbottomelementontheright).Dimensionsinmillimetres;notdrawntoscale.
5.3 Experimentalset‐up
The experiment of a gate section was performed in a 1 m wide and about 90 m longlaboratoryflume.Astraight,verticallyplacedunderflowgate issuspendedinasteel framethatisfixedtotheflume.Figure5.2containsdrawingsoftheplacementofthegateandtheframe in the flume. The dimensions of the gate are 1100x600x50 mm (height x width xthickness); it is a stiffplateand thusacts asa linearmass‐springoscillatorbodywithonedegreeoffreedominthecross‐flowverticaldirection.Topreventmeasurementequipmentand frame parts from influencing the flow, the flume was locally narrowed to 0.5 m byconstructing side walls out of waterproof film‐coated plywood and a sloped ramp of thesamematerialaroundthegate.Inthesectionclosesttothegate,thewallsweremadeoutoftransparentPerspexplastictoallowvisualinspection.Theflowdirectlyupstreamfromthegatewasattachedtothewallsandhaslowturbulenceintensity.
rigid gate body
flow direction
z
D
x
h1
h2
h
102020
20
10
D = 50
6
8
50
82
Figure5.2.Set‐upofgateexperimentintheflume.
310
500
190
600
850
1990
2000
1100
250
250
1000
243
200
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2960
1070
038
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rigid
fram
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flum
e w
all
spin
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to a
djus
t fra
me
heig
htw
ater
tight
bar
rier
slop
e
wat
ertig
ht
plyw
ood
wal
l
sill
slop
e
gate
wat
er
leve
l m
eter
supp
ort b
lock
s
low
er s
uppo
rting
rod
800
low
er s
uppo
rting
rod
low
stif
fnes
s si
de s
prin
g
high
stif
fnes
s le
af s
prin
g
overflow gate
pump, start of flumeplan
vie
w
long
itudi
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-sec
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mai
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rtica
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eter
gate
ope
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sup
porti
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flow
dire
ctio
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outfl
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All
leng
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in m
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wat
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vel m
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Not
dra
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cale
.
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sx
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flum
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83
Figure5.3showsthegate’sfrontviewandsuspension.AsmentionedintheIntroduction,twogate types were tested. The flat rectangular‐shaped bottom (with smooth surface, sharpedgesandwithoutextendinglip)willbecalled‘originalgate’.Theadaptedgatediffersfromthe original gate in that it has five horizontal slots in the upstream face of the bottomsection, as shown in Figure 5.3, and five identical slots in the bottom face of the bottomsection, as shown in Figure 5.4. This gatewill be called ‘ventilated gate’. The slots on theupstreamsideactasinflowopeningsforleakageflowandtheslotsonthedownwardfacingsideactasoutflowopenings.Thedimensionsoftheslotswerechosensuchthattheeffectofthe leakagewould be distinctly perceptible, butwithout compromising the rigidity of thegate.
Figure5.3.Frontview fromupstreamside,showing inflowventilationslots inthebottomsectionofthegate and the three supporting springs.Measurement frame is left out for clarity. Dimensions are inmillimetres.Onlythebottomelementisdrawntoscale.The gate is supported in vertical directionby three springs, one spring in the centerwithadjustablestiffnessandtwosidespringsof lowerstiffness.Priortoeachmeasurement,thegate was set to the desired height and the main central spring was set to the desiredstiffness.Thenthetensioninthetwosidespringswasadjusted(symmetrically)bychangingthelengthofthechainsconnectingthesidespringswiththeframe,seeFigure5.3.Thiswasdoneinsuchawaythatthetwolowstiffnesssidespringscarriedmostofthestaticloadsinvertical direction. The dynamic loads were mostly carried by the stiff central spring. Itsadjustablestiffnessenabledacontrolledvariationofthenaturalfrequency.Forthetwoweakside springs, linear coil springs (Alcomex TR‐1540)were used; for the stiffmain spring adouble leaf springwas custom‐builtwith high yield strength steel and a high elastic limit(Armco17‐7PH,hardeningconditionTH1050).Thetwobendingbladesofthemainspringhavedimensions600mmx30mmx4mm(lxbxt).Thebending lengthL (<600mm)canbeadjustedbymovableblockswithclamps,thusvaryingstructural stiffness, seeFigure5.3.Linearityof thisspringwasconfirmedbystatic
50
600
100 10032 3285
10
gate body
20
bottom element
1100 ventilation slots
side spring
chain force meter
blocks for adjusting stiffness
leaf spring
connection to frame
L
84
loadingtestsfordifferentbendinglengths.Themainspringisinstalledinparallelwithtwosidesprings,eachwithconstantkside=0.57N/mm.TherelationbetweenbendinglengthandtotalstiffnessofthethreespringsisderivedfromconstitutiverelationsplusHookeslaw:
28
1.14, (5.1)
withktotalthecombinedspringstiffnessinN/mm,EthemodulusofelasticityinN/mm2,bthewidth and t the thickness of the leaf spring blades in mm and L the length between theclamps in mm. This formula was calibrated with free vibration tests in air to increaseaccuracybetweenchosenLandachievedktotalanddrynaturalfrequencyf0.Themaingatebodyconsistsofarigidsteelgridfilledwithwater‐resistingfoamandcoveredwiththinplasticplatesonbothsides.ThebottomelementiscarvedoutofPVCmaterialandisscrewedontothemainbody,thusformingonestiffmasswithit.Inordertoreduceweight,thebottomelement ishollow; itcontains fivechambers(seeFigure5.4).Thetotalmassofthe original gatewas 17.2 kg. The bottom element of the ventilated gate is lighter due toremoval of material, but the openings allow more water into the cavities of the bottomelementsothatthetotalmassofthemodifiedgatewasonlyslightlyhigher:17.3kg.Thesevaluesexcludeaddedmassduetowaterdisplacementduringoscillation.
Figure5.4.Planviewofgatewithcrosssectionthroughbottomelement,revealingtheoutflowslotsinthebottomfaceofthegate.Thesupportingframeisleftoutforclarity.Measurementsinmillimetres.Onlythegateisdrawntoscale.AplanviewofthegateissketchedinFigure5.4.Thewaterflowsbetweentheperspexwalls,inthefigurefrombottomtotop.Thespacebetweentheperspexwallsandtheflumewallsisfilledwithstillwaterat thedownstreamwater level.Thesidesealsconsistof thinverticalstripsthatarecarefullyinstalledsuchthatsideleakageisminimizedandatthesametimenocontact is made with the gate. Observations during the experiment indicated that theinevitablesidewaysleakagewasonlysignificantatrelativelyhighhydraulicheads,althoughthe leakage appeared to be only a small fraction of the total gate discharge. It was also
flum
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transparent perspex wall
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upstream
310 190
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ventilation slotsperspex strip
gap ~1 mm
25
30
75
still water at downstream
levelflow
direction
still water at down-stream level
85
observed that thedistancebetweengateand sidewallson thedownstreamsidehad littleimpact on the underflow discharge. Figure 5.4 contains a gate cross‐section through thebottomsection,suchthattheoutflowopeningsoftheventilatedgatebecomevisible.Thesefive openings are identical to the five openings on the upstream face of the gate. All slotswerecutperpendicularlytothefaces,seeFigure5.1.Fivehorizontalsupports, threehingedsteelrodsin longitudinaldirectionandtwoincrossdirection, enableverticalmovementof thegatewhile fixing thegatepositionhorizontally.Sixforcemeterswereinstalled:verticallyoneforeachspringandhorizontallyoneforeachlongitudinalsupport.Onlythemainverticalsupportforceisusedintheanalysisofthegateresponse.Thesamplefrequencywas200Hz.Thelengthoftheanalyseddatafileswas90sonaverageandhadaminimumof60s,yieldingafrequencyresolutionofatleast0.0017Hz.Recording started after reaching equilibrium water levels and horizontal support forces.Figure5.5showsasamplefromananalysedmeasurementsignalofthemeanleafspring.Thesignal analysis was done in MATLAB and consisted of a fast Fourier transforms analysis(FFT).First, the timeaverageof theremainingsteadycomponentof themainspring forcewas subtracted from the signal. Then a slidingwindowwas used to calculate themovingaverage and the amplitude envelopes, which were smoothed with a simple triangularsmoothing function. The Hilbert transform was used to find the envelopes. For eachmeasurement signal, the representative vibration amplitudewas determined as themeandifferencebetweentheenvelopesandthemovingaverage.
Figure5.5.Anexampleexcerptfromameasuredsignalofthemainspring,withcomputedmovingaverageandenvelopesadded.Furthermore,thedischargeandthewaterlevelsontheupstream(h1)andthedownstreamside(h2)ofthegateweremeasuredusingresistance‐typewaterlevelmeters.ThelocationsofthewaterlevelmetersareshowninFigure5.2.SeeErdbrink(2012a)formoredetailsontheexperimentalset‐up.
86
5.4 Definitions
The motion equation in vertical z‐direction for partly submerged bodies includeshydrodynamiccoefficients(cf.Sections2.3‐2.4):
. (5.2)
Here,m ismass, c is damping, k is stiffness,F is the excitation force and the subscriptwindicates the added coefficients. The excitation is a time‐dependent hydraulic forcewhichcanbewrittenas
12
, (5.3)
withWthecross‐flowwidthofthegatesectiononwhichFworks,F0astationaryreferencefluid force and CF is a periodically varying force coefficient. In this chapter not thedisplacementzbuttheresponseforceFzismeasured,theamplitudeofwhichisdenoted .Furthermore, for the pressure on the gate bottom boundary pbound we use the pressurecoefficient CP defined logically as pbound divided by ρg∆h. The two‐dimensional dischargeformulaforanunderflowgatesectioninsubmergedflowis
2 ∆ , (5.4)
with q the discharge per unit width in m3/s/m or m2/s, CD the dimensionless dischargecoefficientforsubmergedflow,atheliftingheightorgateopening,Utheflowvelocityinthevena contracta, estimatedwithBernoulli’s formula, inm/s and∆ .The reducedvelocityVrisusedasdimensionlessdescriptivequantityfortheflow‐inducedvibrations.Itisdefinedhereas
2 ∆
, (5.5)
where fz is the dominant response frequency, the numerator represents the characteristicflow velocity and the gate thickness in flow direction D (see Figure 5.1) is taken ascharacteristic lengthscale.To find theaddedmassmw, it ishereestimatedexperimentallyfromfreevibrationtests inairandstillwater. It follows fromtheratioof thewetanddrynaturalundampedfrequencies.Forf0,waterwehave
,12
, (5.6)
andtheratioisthus
87
, 1 ⁄
1 ⁄. (5.7)
becausetheaddedrigidity(hydraulicstiffness)kwduetobuoyancyonthesubmergedpartofthe gate is found via Hooke’s law: , . The error made by neglectingdampinghereislessthan2%.Numerically,mwnearawallinstillwaterwascomputedinapotentialflowmodelwithafinitedifferencemethodbyKolkman(1988),andstudiedmanytimessincetheninthecontextofgates(e.g.Anamietal.,2012).AveryuniversalapproachistheobservationinKolkman(1984)thatthetotalkineticenergyofthefluidcanbeexpressedas
12
. (5.8)
Summingthevelocitymagnitudeoverallcomputationalnodesshouldyieldavalueformw,assumingthattheobjectvelocityisknown.
5.5 Measurementconditionsandvariationofparameters
Themeasurementprocedurewas tovaryVr anda/D,whichwasdonemostlybychangingsuspensionstiffnessandgateheightandoccasionallybychangingthepumpdischargeandheightoftheoutflowweiroftheflume.Thepresentstudydealswiththecriticalgateopeningrange for cross‐flowvibrations:98%of themeasurements lie in the interval0.48≤a/D ≤1.50. During the experiment the stiffness was varied between and 19.3 N/mm and 967N/mm,correspondingtoarangeinachievedundampednaturalfrequencyinairof5.33Hz< < 37.7 Hz for the original gate with closed bottom section. The gate submergence
⁄ ,withh2 thewaterdepthmeasureddownstream fromthegatewas in therange 4.2 < Cs < 6.2, for 98% of the data. Thismeans flow conditionswere close to fullysubmergedwithminor freesurface fluctuations.DischargecoefficientCD isestimated fromthemeasuredpumpdischarge to beon average0.80with standarddeviation0.11 for theoriginalgateandonaverage0.83withstandarddeviation0.10fortheventilatedgate.Theachieved Vr ranges were 1.2 < Vr < 11.6 for the original gate and 1.8 < Vr < 9.5 for themodifiedgate.TheReynoldsnumberdefinedasRe=UD/ν,againwithU=(2g∆h)0.5,wasinthe range 3.2·104 < Re < 1.3·105. The mass ratio, defined as / , iscomputedfortheoriginalgateas12.3≤mr≤12.7,andfortheventilatedgateas12.3≤mr≤12.8.Dampingwasmonitoredthroughouttheexperimentin32freevibrationtestsinstillwater,wherethegatewassetinmotionbyamanualtaponthetop.Foreachtest,thelogarithmicdecrementofdamping wasdeterminedoverthefirsttenperiods.Atsmalldampinglevels,we have 2 . This formula is used to compute values of the damping ratio . For theoriginal gate, was on average 0.013 and had a standard deviation of 0.0065. For theventilatedgate, theaveragevalueof was0.020withastandarddeviationof0.0061.Thedampingvaluescontainnotrendsrelatedtowaterdepthorstiffnessanddidnotchangeovertime during the experiment. However, there is some inherent variability caused by themanualexcitationforcenotbeingthesameeachtime.Also,itmakesadifferencewhichpart
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ofthedecayingfreevibrationisusedfordeterminingδ.Theventilatedgateshowedasmallnumber of deviant values of higher damping at small openings a/D < 0.7, these are notincludedinthegivenranges.Thehighestrecordeddampingratiointhisparticularsituationwas =0.093.Apossibleexplanation is that flowresistance inside thechambersbetweenthe inflow and outflow slots is more pronounced at small gate openings due to higherrelativevelocities.Furthermore,eventhoughdampingwasnothigheratlargerwaterdepthsin still water, the observed leakage through the side seals at high hydraulic heads (thatexistedforasmallpartofthemeasurements)mayhavehadasignificanteffectondampingduetoskinplatefriction.Thiswasnotquantifiable,sincethefreevibrationtestswereonlyfeasibleinstagnantwater.ThedimensionlessScrutonnumbercombines themassanddamping ratiosand is, for lowdamping,definedbySc 2 4 .Itwasfoundthatfortheoriginalgate1.0≤Sc≤2.3andfortheventilatedgate0.9≤Sc≤3.7.ThevariationintheScrutonnumberisalmostcompletelyduetothediscussedvariationindamping.TheoutliersfortheventilatedgateatsmallopeningsarenotincludedintheScranges.Table5.1givesadditionalinformationonmeasurementconditions.Table5.1.Flumesettingsduringmeasurements.
min max unit
pumpdischargeQpump 13.0 50.0 l/sgateopeninga 22.5 100 mmupstreamwaterdepthh1 0.297 0.656 mdownstreamwaterdepthh2 0.120 0.352 mheaddifference∆h 0.063 0.355 m
Parameter ranges given in Table 1 are based on the combined data set for both gates,consistingof145measurementsfortheoriginalgateplus85measurementsforthemodifiedgate.Theachievedunderflowdischargewassomewhatlowerthanthepumpdischargeasaresult of the sideways leakage through the seals. An observed variation in the pumpdischargeof+/‐0.5l/swasoflittleinfluencesincethefrequencyofthisvariationwasverysmallcomparedtosignalrecordinglength.
5.6 Resultsofphysicalexperiment
The focus of the experimental data analysis is on determining how dominant forceamplitudes in cross‐flowdirection changewithVr for both gate types. Absolutemaximumforceamplitudesdependonstructuraldampingofaparticularset‐upandareoflessinterest;thereforeresponseamplitudesarepresentedfordifferentsettingsrelativetothestationaryhydrodynamicforceF0.Figure 5.6 shows the dimensionless dynamic force response of the closed gate and theventilated gate. Judging from the plot, the response may be divided into three differentregions: 2 <Vr<3 (relatively high stiffness), 3 <Vr<8.5 (medium stiffness) andVr > 8.5(relatively low stiffness). Response values of 5 represent significant, regularoscillations with response frequencies in the range 4.6 Hz < fresp,z < 20.2 Hz. Verticaldisplacementamplitudesestimatedfromtheforceamplitudeas ⁄ wereoverall less
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than0.1D.ThestrongestrecordedforceamplitudeinthehighstiffnessregionwasfoundatVr=2.54.ThemaximumresponseintherelativelylowstiffnessvibrationregionoccurredatVr =10.16.The excitationmechanismsassociatedwith these two regions arediscussed insection4.1.
Figure 5.6. Overview of vibration characteristics of the original rectangular gate (left) and the newventilatedgate (right): reducedvelocityVrversusdimensionlessvibrationamplitude ⁄ of themainverticalforcemeter;aisthegateopening,gatethicknessDis50mm.Theresultsshowthatthevibrationsfoundat lowVroccurquitesuddenly.Therearesteepincreases in force amplitude around Vr = 2 and Vr = 3.0–3.5; most significantly for gateopeningslessthanorequaltoD.Testsatgateopeningssmallerthan0.5Dwerehinderedbytheriskofthegatehittingtheflumebottom.Althoughthegateopeningwasnotvariedovera
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largerange, thedataseemstoshowthatthe forceresponseatVr≈2.5occursatasmallergateopeningthanfortheresponsemaximumatVr≈10.Zoominginonthedenselysampledareaof relativelyhigh frequencyresponseat2<Vr<3gives further insights (Figure5.7).Themeasurementsweredonebymakingseriesofabouttendatapointsofdifferentstiffnesssettings,keepinggateopeninganddischargeconstant.
Figure5.7.Themostsignificantvibrationsat1.5<Vr<4.Measurementsseriesrepresentconstantgateopening a/D and head difference ∆h for varying suspension stiffness. Gate submergence relative todownstreamlevelisdenotedCs.Thesolidtrianglesmarkthegatewithslots,allothersymbolsdenotetheoriginalgate.Responseswith ⁄ 0.25notbelongingtotheseseriesareleftoutforclarity.Theseresultsreconfirmthatinthisreducedvelocityregionsignificantcross‐flowvibrationsoccuratgateopenings intherange0.5D≤a≤D.For1.5<Vr<4,withthestrongestforceamplitudesarounda/D=0.5.FromFigure5.7itappearsthatforheaddifferences∆h≈1.5D–1.75D, vibrationmaximadecreasewith increasinggateopening.Theserieswithhighheaddifference∆h=283mm≈5.7Ddiffersfromtherestbyitshigherhydraulicheadandhigherstiffness settings. Because the still water damping tests proved that damping is notsignificantly higher for high spring stiffness settings, the most likely explanation of thesmallerresponseofthisseriesisanincreaseddampingassociatedwiththehigherhydraulichead.Thisispossiblyrelatedtotheobservedleakagethroughthesidesealsathighhydraulichead.Thisfindingdoesnotinfluencethecomparisonsbetweenthetwogatetypes.However,thereportedhigherdamping for theventilatedgateata/D<0.7, seeSection2.3, suggeststhattheresponsesoftheventilatedgateseriesata/D=0.52areunderestimated.Despite the smaller data set size of the new gate design, the overall effect of applyingventilation holes seems clear. For very similar conditions at 1.5 < Vr < 3.5, the forceamplitudeofthevibrationisafactor3lessfortherectangulargatewithaddedholes.For3.5<Vr <9.5, theholedgate showsavery lowresponse for similar conditions (openingsandheaddifference)astheoriginalgate.FromtheavailabledatapointsatVr>8.5,thepictureoftheeffectofaddingholesisincomplete.MoredataisneededforVr≥10.Theavailabledatasuggeststhattheventilationsucceedsinhavingamitigatingeffectonthevibrationsinthatrangeaswell.
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TheventilatedgateshowsanoverallshifttowardslowerachievedVr‐valuescomparedtotheoriginal gate. The maximum force response of the new gate lies at Vr = 2.19. Also, thismaximum is found at a higher gate opening than themaximum of the closed gate (0.74Dagainst0.48D).Thisistheneteffectofthealteredgatedesign:aslightlydifferentmassandadditionaldischargecausedbythejetofrelativelyhighvelocitythroughthegate.
5.7 Comparisonwithotherexperimentalresults
The measurement data of the original rectangular gate resembles results from previousexperimentalresearch.Itisimportanttorealisethatmostotherstudiesmeasureresponseindisplacement insteadof force. In thehigh stiffnessatVr<5,whichhasbeen studiedmostfrequently, thereducedvelocityatwhichhighestamplitude is found liesclose toVr =2.75foundbyHardwick(1974)andVr,nat=2.5 foundbyThangandNaudascher (1986a,b).ThereducedvelocitybasedonthenaturalfrequencyVr,natissomewhatlowerthantheVrusedinthis study.Thegateopeningsofmaximumamplitudewerea/D =0.67anda/D=0.65 forthesetwostudies,slightlyhigherthanwhatisfoundhere.BilleterandStaubli(2000)reportmaximumforcecoefficientsinz‐directionatVr=2.75andVr=3.0.TheexperimentsofVrijer(1979)coveredvibrationsintherange10<Vr<80.Theirmodeldimensionsweresmaller(gatewidthof10mm).Acomparisonbasedonthesmalloverlapofless than ten data points suggests that the lower bound of vibrations in the low stiffnessregioninthepresentstudyisataslightlylowerVrvalue(Vr=9.0‐9.5)thaninVrijer(1979)whereamplitudesstarttoincreaseforVr>10.
5.8 Summary
Thisexperimentalstudyinvestigatesanewwaytoreducecross‐flowvibrationsofhydraulicgateswith underflow. A rectangular gate section placed in a flumewas given freedom tovibrate in the vertical direction. Horizontal slots in the gate bottom enabled leakage flowthroughthegatetoentertheareadirectlyunderthegatewhichisknowntoplayakeyrolein most excitation mechanisms. For submerged discharge conditions with small gateopeningstheverticaldynamicsupportforcewasmeasuredinthereducedvelocityrange1.5< Vr < 10.5 for a gate with and without ventilation slots. The leakage flow significantlyreducedvibrations.Thisattenuationwasmostprofoundinthehighstiffnessregionat2<Vr<3.5.
5.9 Photographsfromtheexperiment
Figures5.8‐5.12showphotographstakenduringthephysicalexperiment.
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Figure5.8.Sideviewofmeasurementframeinflume.Flowfromlefttoright.
Figure5.9.Detailofthemeasurementinstallation:gatebody,frameandmainleafspring.
lowstiffnesssidespringtobalancestaticforces
leafspring
mainforcemeter
frame
bendinglength
gatebody
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Figure5.10.Originalclosedflat‐bottomgateprofile(top)andmodifiedgateprofilewithventilationholes(bottom).
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Figure5.11.Close‐upofgatebottomsectionwithoriginalbottomprofileinstalled.Flowfromlefttoright.
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Figure5.12.Adjustingthestiffnessoftheleafspring.
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6 Numericalsimulationstudyofgatevibration5
6.1 Introduction
Physics‐basednumericalsimulationsofgatevibrationsthatgivethedisplacementofthegatein time are inevitably complex and computationally involved. The computational Fluid‐Structure Interaction (FSI)model needs to dealwith very small displacements of varyingfrequencies and should represent the boundary forces accurately (see Chapter 3). Manymodelling simplifications, such as lowering the fluid's viscosity or including only partialinteractionbetweenfluidandsolid,arenotpermittedastheymisrepresentthephysicstoodrasticallysothatexcitationmechanismshavenochancetodevelopcorrectly.Meshesthatdonotmatchupatthefluid‐solidinterfacehavethedisadvantageofrequiringinterpolationfunctions(DeBoer2008)anddedicatedloadtransferschemes(Jaimanetal.,2006).RugonyiandBathe(2001)exploreFEMforFSIproblemswithincompressiblefluidflow.Itwasclarified inpreviouschapters thatstudies flow‐inducedvibrationsofbluffbodies ingeneralusuallyrelatetovortex‐inducedvibrations(VIV)whichdiffers fundamentally fromgatevibrations.Examples for the famousproblemof flowaroundstationaryandvibratingcylindersareAl‐JamalandDalton(2004)andDaietal.(2013).Numericalmodellingofgatesoftenhasthegoalofverifyingthedesignofaspecifichydraulicstructure.Determiningstaticsupport forces is then a greater concern than addressing vibrations (Scheffermann andStockstill,2009;Liuetal.,2011).Thosestudiesthatdoconsidervibrationsmoreoftenthannot fail to make links to the existing state of the art, thus not optimally contributing tofundamentalknowledge(e.g.Lupuleacetal.2007).
6.2 Modelset‐up
Thegoalofthenumericalsimulationsistodescribethephysicalfeaturesthataredecisiveforcausingthedifferenceinresponsebetweenthetwogatetypesthatwereintroducedinthepreviouschapter;astandardrectangulargateandarectangulargatewithventilationslotsadded. The numerical model is two‐dimensional in the vertical direction (2DV), see thefigureinSection5.2.Modeldimensionsareidenticaltothephysicalscalemodel.Theflow‐wise model length is 3.5 m. The static vertical equilibrium is achieved by a suspensionsupport force equal to the difference between gravitational and buoyancy and mean liftforces. The flowvelocity vector isdefined as , . The fluid flow ismodelledby theincompressibleReynolds‐AveragedNavier‐Stokes(RANS)equationsincombinationwiththestandardk‐εturbulencemodel.Thegoverningequationsare:
∙ ∙ ⨂ ∙ ,
5Thischapterisbasedonandusestextandcontentfrom“Reducingcross‐flowvibrationsofunderflowgates: experiments and numerical studies” by C.D. Erdbrink, V.V. Krzhizhanovskaya, P.M.A. Sloot,currentlyunderreviewattheJournalofFluidsandStructures.
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and ∙ 0, (6.1)
whereρisdensityofwater,μisthedynamicviscosityofwater,pisthewaterpressure,barsindicate time‐averaged values and primes indicate variational variables, according to theReynoldsdecomposition.Thegravitytermisaforceperunitvolume:
0
. (6.2)
Thek‐ε turbulencemodelprovidesclosurebysolvingacoupledpairofPDE’sforturbulentkinetic energy k and turbulent dissipation ε (see section 3.3 and COMSOL 2013). For theboundaryconditionsofthefluiddomain,thereisablockvelocityattheinlet:
0
, (6.3)
with U0 a chosen constant. At the outlet a hydrostatic pressure profile is imposed:
,wherehout is thewaterdepthat theoutlet,equal to theheightof theflowdomain.Thebottomflowboundaryis 0,noslipwithawallfunctiondescribingthenear‐wallvelocityprofile.Theboundaryat thewatersurface is ∙ 0, freeslipor ‘rigidlid’,withnthenormalvector.For the initial conditions of the fluid domain, we have at t 0, 0 for all , , and
0 ,withS t asmoothS‐curveincreasingfrom0to1for0 t 0.5s.Thewall function condition for full resolution consists of wall lift‐off dimensionless walldistance of 11.06with:
with shear or friction velocity ⁄ √ . (6.4)
Thisprescribes thedistanceatwhichtheviscoussub‐layerandthe logarithmic layermeet(seee.g.COMSOL,2013).The description of the solid domain includes displacement of the whole body, withdisplacementvectorus,anddeformationduetostresses.Defining , ; , andσsasthestressesitexperiences,wehave
∙ . (6.5)
TheappliedexternalforcesFvactingonthegatebodyare
0 0
, ,, (6.6)
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whereFwater represents thedynamic loadof thewater flowon the submergedpart of thesolid. The spring and damping forces depends on chosen constants for stiffness (k) anddamping(c).Thelastforcerepresentsthesteadysuspensionforceinthestructuralelementthatconnectsthegateobjectwiththebigger(fixed)structure.Theconstitutiverelation,orstress‐strain relation, can bewritten as : ε ε with sub‐indices 0 indicatinginitialvalues,Ctheelasticitytensorrepresentingmaterialproperties(Young’smodulusandPoisson ratio) and : denoting thedouble‐dot tensorproduct.The strain tensor is givenby
1 2⁄ ,whichisequivalentto
, and12
. (6.7)
The displacement of the solid is fixed in horizontal direction: , 0 and the initialconditions of the solid domain are 0 and ⁄ 0. The load that thewater flowforces exerton theboundariesof the solidbody,Fwater, is definedas a forceperunit area
∙ (inN/m2),wherenisthenormalvectoronthesolidboundaryandTisthe2x2tensorcontainingallstressesoftheflow.Thenormalstresses,onthediagonalofT,representthe pressure. The remaining elements ofT represent the viscous shear stresses – each oftheseelementsconsistofastresscontributionfromtheturbulentviscosity(μT)andfromthelaminar viscosity (μ). At t = 0, 0. For 0 < t ≤ 0.5 s the force on the boundaryincreasesaccordingtothesamesigmoidfunctionS(t)asfortheflowvelocity: ∙ ∙
.TheALEmethodisagainusedtomakeamovingmeshfortheentiremodeldomain,seealsoSection 3.3. This means that themesh covers fluid and solid domain together. Themeshadaptstothetranslationalmotionofthegatewithoutchangingitsconnectivity.Theinitiallystraightupperfluidboundary,therigidlid,slightlybendsupanddownalongwiththegate,asitispartoftheALEmesh.Theboundaryconditionsforthemeshdisplacementvectorumare
, 0 on the fluid surface boundaries,
0onallremaining boundaries of the fluid domain,
on the solid boundaries. (6.8)
For the inner area, the mesh deformation is smoothed using ‘hyper‐elastic smoothing’,inspiredbyandescriptionofneo‐Hookeanmaterials,anapproachthatseeksforaminimumof‘meshdeformationenergy’(seeCOMSOL2013).ThissmoothingtypeprovedmorerobustduringthepreliminaryrunsthantheWinslowtypeusedinChapter4.Againthefiniteelementmethod(FEM)isused.Themodeliscalled“modelII”inTable1inSection3.3.3.The transient runsareprecededby steady‐statepre‐runswith thegateheldfixed in order to iteratively determine the inlet velocity at which the upstream pressureaway from the gate is zero at the surface. The grid consists of unstructured triangularelementsand inflatedboundaryrefinementsadjacent to thegatewallsand flowbottom.Atypicalgridcontainedaround35,000elementsand200,000degreesoffreedom.Thecellsize
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inthenear‐wallflowregionsisdictatedbythedimensionlesswalldistancebyensuringthaty+max=11.06atallwallswherewallfunctionsareapplied;thiswascheckedaftereachrun.TheFEMsolverCOMSOLMultiphysicsv4.3a(COMSOL2013)isusedtosolvethesystemofequationsofsevendependentvariables.Thesolutionprocedureissolvedinafullycoupledwaywith the PARDISOdirect solver. The implicit backward differentiation formula (BDF)withadaptivetimesteppingisused,anextensionofthebackwardEulermethodforvariableorder.The simulationsweredoneona cluster, using24 coresona singlenode.Typically,threesecondsofsimulatedtimetookaround12hoursofcomputingtime.
6.3 Selectedcasesandmodelvalidation
Two cases from the experimental data set are selected for simulation; these arerepresentativeofstrongvibrationsinthelowandhighVrregion.SeeTable6.1.Table6.1.Selectedcasesfornumericalsimulations(valuesfromexperiment). Q(l/s) k(N/m) h1(m) ∆h(m) a/D(‐) Vr(‐) Cs(‐) fz(Hz) (N)
case1 23 82502 0.398 0.088 1.00 2.6 5.2 10.1 25.6
case2 40 19298 0.605 0.285 0.86 10.2 5.5 4.66 63.0
Weareinterestedinthebehaviourofbothgatesforconditionsascloseaspossibletothosefound in the experiment. The followed approach was to assign applicable physicalparametersfromtheexperimenttothenumericalmodel.Firstly,theaddedmasscoefficientmw in still water is derived from a zero discharge model and compared to experimentalvalues.Secondly,iterativevalidationrunsaremadeinamodelwithdischargetoachievethesettingsnecessary forattaining the response frequencyas found in the experiment.Thesetwo preparatory modelling steps should be seen as efforts towards model validation(comparing the modelled value of a universal physical parameter with the experimentalvalue) and calibration (adjusting the numerical model for the specific modelling task),respectively.6.3.1 AddedmassvalidationThekineticenergyapproachisfollowedtofindmwinstillwater.Becauseflowvelocitiesarelow,thelaminarincompressibleflowequationsaresolved.Theoscillatinggateissimulatedbyamovingwallwithaprescribedverticalperiodicvelocitywithafrequencyof4.7Hzandamplitude of 0.15m/s. The rigid lid assumption is justified for this situation because thewaveradiationeffectissmall:forh=0.40m,wehave ⁄ 36 ≫ 10,theusualcriterionforwaveradiation(Kolkman,1976).
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Figure 6.1. Added mass coefficient in still water for various openings a and water depths h, fromexperiment(exp)andsimulations(sim).InFigure6.1experimentalvaluesofnon‐dimensionaladdedmassinstillwaterareplottedfor threewater depths, togetherwith simulation results for h = 0.40m. The addedmasssimulationsshowthat thewallproximityeffectat lowa/D issomewhatmorepronouncedthanintheavailableexperimentalvalues.Forvibrationsathighgateopenings,theinfluenceof the bottom disappears. The simulations show a gradient reduction in correspondencewiththis,whileexperimentaldataathighera/Dwouldbeneededtocoverthisstabilisation.ExperimentaldatabyNguyen(1982)inNaudascherandRockwell(1994)showstabilisationof the addedmass for a/D > 2–3. The dependence on water depth that is shown by theexperimentalvaluesisnotsufficientlycapturedbytheFEMmodel: forotherwaterdepths,theresultsaretooclosetotheplottedrelationforh=0.40m.Theinfluenceofdomainwidthwasneutralisedby ignoringvery small flowvelocities in the computation, atdistances faraway from thegate. In summary, for the combinationsofwaterdepthsandgateopeningsthatareconsideredinthisstudy,thephysicaladdedmassisreasonablywellapproximatedbythenumericalmodel.6.3.2 ArtificialaddedcoefficientsIn the numerical model, the gatemass is defined by assigning a solid density ρs, and thesuspensionstiffnesskisdefinedinaspringsuspensionsupport.Theaddedrigiditykwduetobuoyancy is in all studied cases negligibly small, i.e. kw << k. Despite the fact that foroscillating bodies in flowingwatermw andkw are in general not equal to their stillwatervalues(norconstantsforthatmatter),itisreasonedthatmismatchesofsimulatedresponsefrequencieswith experimental values,must be traced back to artificial added coefficients.Suchmisrepresentationswereindeedfound.Itwascheckedthatthenaturalfrequencyoftheisolatedgate invacuumcalculatedby the sameFEMmodel exactlymatched theanalyticalvalue.A possible cause of numerical complications in FSI is the so‐called ‘artificial addedmass’effect. This has been investigated and partially described and explained for sequentially
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staggered schemes (Förster et al., 2007), but no literature was found for FSI with fullycoupled schemes in FEM for relatively stiff solids. Similar to Jamal andDalton (2004),wedefinethemassfactorm*tobetheratioofsoliddensityoverfluiddensity.Theterm“massratio”isalsousedforthissometimesbutcanbeconfusedwiththedimensionlessmassmr.Thesensitivityofthesimulatedresponsefrequencyfz,numtothemassfactorm*isplottedinFigure6.2.Using the true mass factor of the experiment, which is less than one, gives crudeunderestimates of themeasured response frequency fz,exp for both investigated cases. Themass factor is increased tom*=1.1 simplybyusinga lowergateheight in thesimulationthanintheexperiment.Whenm*isincreasedfurtherbymeansofincreasingsoliddensityρs,while at the same time adjusting k proportionally such that the dry natural frequency f0remainedconstant,thenumericalmodelgivesbetterestimatesoffz,exp.Gridrefinementsdidnotnoticeablyinfluencethetrends.
Figure6.2.TheartificialaddedmasseffectforthetwocasesofTable1.Thesimulatedfrequenciesfz,numaregivenfordifferentmassfactorsm*.Forbothcases,alldatapointshavethesametheoreticalnaturalfrequency.Thehorizontallinesshowtheexperimentalresponsefrequenciesfz,exp.Increasingm*atfixedf0doesnotyieldanasymptoticapproachtotheexperimentalvalues,however. In addition, the achieved displacement amplitudes of the gate vibration areinconvenientlysmall form*≥10(order0.1mmandsmaller).Asatradeoff it isdecidedtotakem*=3andadaptkaccordinglysothatnatural frequencyandresponsefrequencyarewellrepresentedbythenumericalmodel forbothcasesanddisplacementsare inorderofmillimeters. It is noted that this measure distorts the absolute values of the modelleddisplacements.
6.4 Resultsofcalibratednumericalmodel
Next,cases1and2weresimulatedwiththecalibratednumericalmodel. Ineachtransientrun, only three to four seconds were simulated. The numerical perturbations related toinitialisationoftherunwereenoughtokick‐startavibration–thenthesimulatedvibration
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eitherquicklygrewinamplitudeordampedout.Afterasubstantialnumberoftestruns,noactiveresponsehasbeenfoundforcase1forneithergatetype.Thisispresumablyrelatedtoanunderestimationofthedynamicsoftheimpingingshearlayer,seeSection4.1.Forcase2growingamplitudeswerefoundataslightlylargergateopeningthanintheexperiment(46mm in the simulation versus 43 mm in the experiment). All plots in this section aresimulationsofcase2;subsequentanalysisalsofocusesoncase2.Moviesofsimulationcase2in theonlinematerial showpressure, velocityand turbulent kinetic energy fields forbothgates.Figure6.3showstimesignalsofthesimulateddisplacementnormalizedtothegatewidthD.Afteraninitiallysimilardisturbance,theventilatedgatefollowsadampedvibrationandtheoriginalgateisamplified.Thefrequencyoftheventilatedgateisalittlelowerthantheclosedgate.
Figure6.3.Simulatedverticaldisplacementsoforiginal,closedgate(thincontinuousline)andgatewithventilation slots added (thick dashed line). The displacement is plotted relative to the original gatepositionz=aandnormalizedtothegatewidthD.Zooming in on the gate response, Figure 6.4 provides a closer look on the displacementsignal,givingalsoaccelerationandliftforceoscillation.
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Figure 6.4. Simulation case 2: time signal excerpts of gate displacement, gate acceleration anddimensionless lift force, for both gate types. The acceleration is computed with a first‐order finite‐differenceschemewithoutfiltering.For further comparison of themodelled response of both gate types, a number of profileplots is made in Figures 6.6‐6.10. Figure 6.5 gives the locations of output profiles forpressure,velocityandturbulentkineticenergy,whichareplottedinFigures6.6‐6.10.
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Figure6.5.Locationofoutputprofiles(dashedlines)usedinFigures6.6‐6.10.IneachplotofFigures6.6‐6.10,theninethinlinescorrespondtodifferentmomentsinonefull sineperiod, according to π 4⁄ with0≤ i ≤8, starting inequilibriumposition (movingupwards)att=1.35s.Fortheoriginalgate,theivaluesareindicatedexplicitly.Additionally,Figures6‐10displaythesituationwherethegateisfixedattheequilibriumpositionforbothgatetypes.Thevariationofflowparametersismuchsmallerinthefixedgatescenarioandthereforethesearedepictedassinglethickdashedlines,whichdenotetime‐averagedvaluesover twoconsecutiveperiods, after foursecondsof simulation.Figures6.6‐6.10 showthatthe oscillating ventilated gate experiences significantly lower periodic variation instreamwisepressure,velocityandturbulentkineticenergy(TKE)atthetrailingedge,andinbottompressure,thanthegatewithoutholes.Obviouslythisisassociatedwiththedifferenceindisplacementamplitudeatthetimeofoutput,butneverthelessaninsightfulcomparisoncanbemade.Thestreamwisepressureplot(Figure6.6)showsthat,ataheightofhalf thegateopening,the original gate experiences larger temporal variation and higher maximum streamwisepressuregradientsdirectlyunderthegate(0<x/D<1)thantheventilatedgate.Thelatterexperienceslow|dp/dx|‐valuesforx/D<2.Theu‐velocityprofiles(Figure6.7)clearlyshowthe influence of the extra stream through the hole. For the ventilated gate the area ofnegativestreamwisevelocityisverylimited,0.8<z/a<1versus0.7<z/a<1fortheoriginalgate,andthenegativeu‐velocitiesaremuchsmalleraswell.
zx
D
4D
a/2
4Da
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Figure6.6.Simulatedstreamwisepressurevariationatz=a/2 formovinggatewithout(left)andwithventilationholes(right);phydristhehydrostaticpressureatz=a/2.Thickdashedlinesshowthesituationwhere the gate is held fixed. Vertical thin dashed lines indicate location of the gate. The nine linescorrespondtodifferentmomentsinonefullperiod,accordingtoiπ/4with0≤i≤8.
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Figure6.7.Simulatedverticalprofileofhorizontalvelocitycomponentuatdownstreamgateedge.(frombottom togate).Thickdashed lines indicate the fixedgate scenario.Positive values indicate themeanstreamwisedirection.
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Figure6.8.Simulated verticalprofileof vertical velocity component vatdownstreamgate edge. (frombottomtogate).Thickdashedlinesindicatefixedgatescenario.Positivevaluesindicateupwarddirection.
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Figure6.9.Turbulentkineticenergyinverticalprofileatthetrailingedgeofthegate.Thickdashedlinesindicatefixedgatescenario.
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Figure6.10.Spatialvariationofhydrodynamicpressureonthegatebottomboundaryforbothgatetypes.Thick dashed lines indicate fixed gate scenario. The vertical dashed lines show the location of theventilationslot.
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Theeffectofthedownwardjetthroughtheholeonv‐velocity(Figure6.8)isthattheoutflowdirectionthroughtheslotisnowalmostcompletelydownwardduringthewholeoscillationperiod,whereas significant upward velocities exist for the original gate. The TKE plots inFigure6.9showthattheTKEmaximumisgreatlyreduced.Intheventilatedcase,thereareno large TKE levels close to the gate and maximum levels occur around z = 0.75a. TheasymmetricshapeoftheTKEprofileoftheventilatedgateisduetothecombinedpresenceof two shear layers, resulting from flow separation from the downstream edgeof the slotoutlet (upperpartofprofile) and from the leadingedgeof thegatebottom(lowerpartofprofile).Theboundarypressures inFigure6.10aregivenby thepressureparameterCp,definedasthetotalpressuredividedby ∆ .Theplotsindicatethatthereducedtemporalfluctuationforthenewgateresultsinpositivepressuresduringthewholecycle,whereastheboundarypressureontheoriginalgateislargelynegativeinthesecondpartofthecyclewhenthegateisbetweenlowestandmidwayposition.Thespatialpressurevariationstowardsthetrailingedgeoftheoriginalgateseemtosignifyunstableflowfluctuations.Thesearevirtuallyabsentattheslottedgate,forwhichasmoothpressuredropisvisibleacrosstheoutflowfromtheventilationslot.Figure6.11givesmoresimulationresultsofbothgatetypesfordifferentmoments intimeforoneoscillationperiodofcase2.Theflowfieldsoftheoriginalgateareoutlinedontheleftand;theventilatedisontheright.Section6.8showssimilarplotsforcase1fortheoriginalgatetype.
Case2:originalgate,fz=4.7Hz Case2:ventilatedgate,fz=4.7Hz
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Figure6.11.Flowvelocityvectorsandpressurefieldsforthesimulatedmotionofbothgatetypesforcase2(f=4.7Hz).Theplotsarenumberedaccordingtothetimestepsofonefullperiodasindicatedinthetopplots.TheplotsofpressurecontourlinesandflowvelocityvectorsinFigure6.11servetopointoutqualitative differences between the two gate types. Overall, the modified gate withventilation slots has a flow field that is more constant in time with less vena contractavariationandlesswakefluctuations.Itcanbeseenthattheflowthroughtheslotworksasanextraseparatedshear layerandalsokeeps instabilities fromthewakeawayfromthegate,seeplots3and4.Atthistime,duringthefirstpartoftheupwardmovement,thereversionofstreamwiseflowvelocityatthebottomboundary(whichispartofapermanentrecirculationzone)isstrongest.Instantaneousspatialpressuregradientsnearthetrailingedgeappearforthegatewithoutholesinplots3and4.Forthegatewithslotsthisfeatureisnotvisible.ThisisdiscussedmoredeeplyinSection4.1inthecontextofexcitation.Thejetthroughtheholepersists in its downward flow during thewhole oscillation period. This causes a relief inpressure build‐up and prevents pressure fluctuations from the wake to enter the regionunderthegate.Theintersectionsofcontourpressurelineswiththegatebottompresentintheplotsoftheoriginalgatehelptoillustratethedifferenceinpressurevariationsbetweenthetwogatetypes.ComparingFigure6.11withthesimulationsofcase1fortheoriginalgateinSection6.8,itisimmediatelyclearthatcase1hasaconsiderablymorestableflowfield.Thevenacontracta,thelargewakerecirculationandthesmallrecirculationzoneunderthegatehardlyvaryintime.
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originalgateinfixedposition
ventilatedgateinfixedposition
Figure 6.12. Flow velocity vectors for the situationwhere the gate is fixed. Boundary conditions areaccording to case 2. The gate position corresponds to the equilibrium position of the moving gatesimulationsasgiveninFigure6.11.Additional transient simulations with the gate held fixed (Figure 6.12) again show thestronger reversed flow along the gate bottom for the gate without slots. There is alsoevidence for flow entering the near‐bottom region from downstream, similar to plot 4 ofFigure 6.11. This entrainment feature is discussed in the next section. Table 6.2 gives themaximum flow velocities for the simulations. Velocities are much higher in the situationwhere the gate is free to move. There is little difference between the two gate typesregardingmaximumvelocity.Forthemovinggatescenariothemaximawereattainedduringthefirstpartofthedownwardmovementofthegate.Table6.2.Maximumsimulatedflowvelocitiesforcase2.
movinggate fixedgate
originalgate 3.38m/s 2.38m/s
ventilatedgate 3.30m/s 2.40m/sAn additional series of numerical simulations was made with a different ventilation slotdesign:itlacksahollowchamberandtheventilationslotnowmakesa45degreesanglewiththegatesides.Theflowfieldplotsforconditionsequivalenttocase2areincludedinSection6.9. These simulations show quite similar flow characteristics as the ventilated gate typediscussedthusfar.Thereisnowakeentrainmentduringthevibration(plot4ofFigure6.14),
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norwhen the gate is fixed (Figure 6.15).Whether the higheru‐velocity at the ventilationoutflow,duetotheoutflowanglebeingtiltedmoreinstreamwisedirection,isbeneficial, isinconclusivefromthesetests.Complicatingfactorincomparingdifferentgatedesignsisthatthedischargecoefficientsvaryandthereforeotherupstreamvelocitiesarerequiredtoreachthe same flow conditions, and moreover that different masses give different submergednaturalfrequenciesandhencedifferentVrvalues.
6.5 CombineddiscussionofphysicalexperimentandFEMmodelling
6.5.1 VibrationmechanismAsexplainedintheintroduction,pastresearchlargelyagreesontheexistenceandworkingoftwoexcitationmechanismsforcross‐flowgatevibrations:ILEVandMIE(noiseexcitationor Extraneously Induced Excitation (EIE) is not considered here). Based on literature, theinstability‐induced or vortex‐excited type is themost likelymechanism for the vibrationsfound around Vr ≈ 2.5, while the movement‐induced type is probably the dominantmechanismforthevibrationsatVr≈9.5.Letusconsiderthefluid‐structuredynamicsoftheoriginalgate.Theseparationpointisfixedatthesharp leadingedge– irrespectiveofdischargeandgateposition–andtheboundarypressure fluctuatesnear the trailingedge,both in spaceand time.However,given the factthatthestrongestvibrationsintheexperimentoccurredforconditionssimilartocase1(viz.atVr≈2.5),itisremarkablethatthesimulationofcase1vibrationsdampedout.Runswithadapted settings in the neighborhoud of case 1 (higher or lower gate opening, higher orlowerdischarge)wereattempted,butdidnotyielddifferentresults.Atransientsimulationofcase1conditionswiththegateheldfixedshowedonlysmallflowinstabilitiesintheshearlayer, making it impossible to determine the Strouhal number and identify small‐scalevortices.Simulationcase2displaysthemain featureofmovement‐inducedexcitation(MIE)orself‐excitation, because not only is a motion required to initiate the vibration process, thefluctuationsintheboundaryexcitationforcedonotappearatallinthetransientsimulationwhenthegatepositionisfixed.Thefactthatthephaseshift betweenexcitationforceanddisplacementisclosetozeroandthemeanlift isonlynegative foraquarterof theperiod,impliesthatthereisnodirectevidenceofgalloping‐typeMIEexcitation(BilleterandStaubli,2000).Fromthevectorplotsoftheoriginalgate(case2),itisapparentthatadelayexistsbetweenthemotionofthegateandtheoscillatingshearlayer.Duringthedownwardgatemovement(plots 1‐3 in Figure 6.11), the reduction of the vena contracta lags behind the gatedisplacement,whichresultsinaminimalverticaldistancebetweentheregionofseparatedhighvelocityflowandthetrailingedgeatt=2.125s,atthelowestgateposition.Theprofileplotofv‐velocityinFigure6.8showsthatthereismildupwardflowatx=Datthispartofthe period (6π/4 to 7π/4or i = 6, 7), but it occurs away from the gate,with strongest v‐velocitiesatz=0.5a–0.7a.Asthegatemovesupwardsthroughtheequilibriumposition(plot4ofFigure6.11,att=2.175s),thevenacontractaattainsitsminimum.Thisleavesasizeablegapbetweenseparatedflowandgatebottom.Clearlyvisiblefromthesameplot,entrainmentfrom the wake into the recirculation zone directly under the gate leads to high local
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velocitiesaroundthetrailingedge.ForashorttimethereisasmallregionofrelativelylowpressureatthetrailingedgeandthereverseflowcomingfromthewakeseemstoseparatefromthetrailingedgeasitmovesfromrighttoleftinFigure6.11,plot4.ThisphenomenonisreflectedintheprofileplotsinFigure6.7bynegativeu‐velocitiesat0.8a<z<afori=7,8,0,1.ThehighestTKElevelsarefoundatthesamepartoftheperiod,seeFigure6.9.Thegateappearstoreceiveasignificantupwardpushfromthepressurefluctuationsassociatedwiththeentrainment.Ifthishappensrightatthismomentoftheperiod(t=2.175s,timestep4inFigure6.11), as theresultssuggest, then thispushdoespositiveworkon thegatemotion,sinceitisinphasewiththegatevelocity.Thevectorplotsuggeststhatthispushtakesplaceatx=0.6D–0.7D.However,theboundarypressureplot(Figure6.10)containssmalldentsfori=7,8,0,1aroundx=0.9D,indicatingthatitmighthappenclosertothetrailingedge.Themovies of the original gate in the supplementary on‐line material provide further visualsupportofthisexcitationprocess.Kolkman and Vrijer (1987) reported on a mechanism based on streamwise flow inertiacausedbythedischargenotinstantlyadaptingtothenewgatepositionduringlowfrequencyvibrations at small gateopenings.Because thedischarge isnever choked in thenumericalmodel,thismechanismisbydefinitionnotsimulated.However,theroleofthismechanismcannoteasilybeputaside,judgingfromtheprominentstreamwisepressureirregularitiesinthesimulations.6.5.2 EffectofflowthroughtheventilatedgateComparing the described entrainment effect of the original flat‐bottom gate to the flowdynamicsoftheventilatedgate,asplottedinFigure6.11, it isseenthatthevenacontractadoesnotexperiencethesameoscillatoryvariation.Thefreeshearlayerandthehighvelocityseparatedflowmaintainaconstant,safedistancefromthegatebottom.Moreover,therearenosignsofwakeentrainmentfortheventilatedgate,sothatanexcitationbythismechanismmust be absent. The profile plots in Figures 6–10 demonstrate the lack of near‐bottompressurefluctuationscapableofexcitationbythevirtualabsenceofnegativeu‐velocitiesandpositivev‐velocitiesandmuchsmallerTKElevels.Theflowfieldsforthefixedgates(Figure6.12) illustratethedifferencesbetweenthetwogatetypesaswell: theoriginalgateshowsreverse flow near the trailing edge and a significant recirculation bubble under the gate,whilethetworecirculationzonesundertheventilatedgate(thatarevisibleonlyfromcloseinspection)aresmallerinsizeandhavenegligiblysmallvelocities.FortheMIEmechanism,thegeneralworkingoftheleakageflowthroughtheventilatedgateis that it alleviates the fluctuation of the streamwise pressure and the fluctuation of theboundary pressure (thus reducing or removing negative lift). In particular, it works bypreventing wake fluctuations from entering the near‐gate region. The simulations wereunfortunatelyunabletocapturetheeffectofaddedslotsinthecaseofvibrationduetoILEV.Itisplausible,however,thattheleakageflowinthiscasedisruptstheseparatedshearlayerandvorticesshed fromthe leadingedge.Thedownward flowwouldconsequentlyremoveinstabilitiesfromthegatebottomandhencereducetheprobabilityandimpactofunstablereattachment.The jet that flows out the middle of the bottom boundary not only removes localdisturbancesfromthevicinityofthegate,aswehaveseen,butalsohastheeffectofsplitting
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theflatbottomsurfaceintotwopartsasifthereweretwothingates.Thisisfavourablesincethinnerprofilesarelesspronetovibrations,oratleastreducetheriskygateopeningregion.An alternative explanation of the response reduction could be found in the fact that thepresence of the slot simply decreases the surface area for the lift pressure to work on.However,simulationsofaclosedrectangulargatewithabottomsurfaceareaequivalenttothatofthegatewithaholeshowedthatthediminisheddynamicforceamplitudecannotbeexplained by the smaller area. The different response must therefore be caused by theleakageflow.The location of the outflow openings and the angle atwhich the jets flow (relative to thegate’s bottom surface and themean flow) are undoubtedly important design parameters.Another factor is thewidthwisedistributionof theslots in thedesign, that is, thedistancebetweenconsecutiveopenings.Ajetoriginatingfromflowthroughasmallcircularopeninginto a wide domain expands in all three dimensions, this is related to entrainment ofsurroundingfluidofrelativelylowvelocities,aninherentlyturbulentprocess.Thesituationofflowthroughhorizontalslotsinthepresentstudyisdifferentfromthecanonicaljet,sincethe openings are elongated in one direction and the outflow domain has strong velocitygradients. The spaces between the slots, that were necessary in the present design topreservestructuralrigidity,influencetheoutflowsfromthegatebottomslotsandthethree‐dimensionalaspectoftheflow.So,althoughtheslotconfigurationinthisstudyissymmetricaroundthemiddleofthegate,three‐dimensionalflowphenomenamayplayanon‐negligiblerole.Theimpactofthisongatestabilityandtheeffectivenessoftheleakageflowcanonlybeclarifiedthroughadditionalphysicalandnumericalmodelling.6.5.3 ImplementationinprototypegatesInprinciple,thereisnoreasonwhytheventilatedgatedesignwouldnotbeapplicabletoanactualgate.Thegatewouldfunctionasaconventionalgatemostofthetime,withanaddedleakflowdischargethroughtheventilationslots.Perhapsthemostcriticalissueisthegate’sbehaviouratverysmallgateopenings,directlyprior to full closure. Inclosedposition, theleakageflowthroughtheslotsmustbezero.Thisrequiresgoodsealsthatatthesametimeshouldnotdeteriorate thevibrationpropertiesof thegate’sbottomgeometry.The idea toadd amovable element (gate or valve) to control the inflow through the upstream inflowslotsseemsinteresting,butofcoursemightengageitselfinflow‐inducedvibrations.Anotherpoint of attentionwould be to prevent the accumulationof sediment and trash inside theventilationholes.6.5.4 EvaluationofnumericalmodellingThe rigid‐lid approach makes it impossible for the model to capture the effect of waveradiation. This is a form of hydraulic damping that involves vibration energy to betransformed irreversibly into free‐surface waves. Not modelling this is not a severeomission,seealsosection3.2.Intheexperimentwaveradiationwasonlyobservedonceattheupstreamwatersurfaceinthelowstiffnessvibrationregion.Anotherfree‐surfaceeffectthat is completely neglected by using the rigid lid approximation is the coupling of flow‐induced undulations of the downstream surface with the gate oscillation at increasingFroude number (see Naudascher and Rockwell, 1994). However, these undulations onlyoccuratmuch lowersubmergence levels than investigated in thisstudy.Aconsequenceofthe fixed free surface boundary in connectionwith the constant discharge is furthermore
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that global pressure oscillations are exaggerated. A lowering of the gate results in animmediatepressure increase in theupstreamregion, for instance. It isnot easily found towhat extent this affects the emergence of gate instabilities in the simulations; there ispresumably a linkwith the representation of theMIE vibrations. The complication of thisnumericalmodelling aspect lies in the fact thatmethods that do simulate the free surface(suchasVolumeOfFluidorPhaseField)areusuallynotsuitableforincludingbothmovingobjectsandturbulentflow,ortheyarecomputationallyimpracticallyexpensive.In physical experiments, comparisons ofmeasured gate displacement and excitation forcecannotalwaysdistinguishbetweenaddedmassandrigiditysincebothcoefficientsarepartoftermsin‐phasewiththedisplacement(e.g.Kolkman1984).Similarly,inanumericalstudyit is not sufficiently clearwhether a deviating response frequency should be attributed toartificial mass or rigidity effects. Either way, the distorting influence of artificial addedcoefficientsmust be elucidated before the fullworking of all physicalmechanisms can beuncoveredbynumericalmodelling.Artificialdampingornumericaldiffusion canbe tracedback to theuseofan implicit timescheme and ‘consistent stabilisors’ (COMSOL 2013). The effect of this is inherent to thefollowednumericalapproach;withoutitthesimulationoftheflowfails.Naturally,toomuchartificialdampingofturbulentflowresultsinvibrationsnotbeinginducedinthesimulationswhere they do occur in real life. In particular, negative damping plays a key role in self‐excited vibrations – if negative damping is neutralised by artificial factors, MIE‐typevibrationswillbeinaccuratelyrepresented.AsmentionedinSection4.1,anotherlimitationofthenumericalmodelistheconstantdischargethatinreallifefluctuatesasaresultofgatedisplacements(mostprominentlyforlowfrequencyoscillationsatsmallgateopenings).Finally,theobviouslimitationoftheRANSapproachwithaturbulencemodelisthatnotallturbulence scales are simulated. The fact that vortex shedding and velocity and pressurefluctuations at small‐length scales connected to the shear layer are parametrised toocoarsely could be the reason why the (presumably ILEV‐dominated) vibrations of case 1were not reproduced. Applying a different turbulence model is unlikely to improve this;LargeEddySimulation(LES)couldbetheonlywayforward.
6.6 CombinedconclusionsofphysicalexperimentandFEMmodelling
Thegoalofthisthethischapterandthepreviousonewastoexperimentallyandnumericallytest a new hydraulic gate design for reducing flow‐induced cross‐flow vibrations. Theadditionof ventilation slots toa rectangular flat‐bottomgatealloweda controlled leakageflow through the bottom of the gate, which produced a less severe vibration responsecomparedtoanunalteredreferencegate.Theexperimentaldatasetfullycoversthetransitionsbetweenconditionswithandwithoutsignificantflow‐inducedvibrationsinthereducedvelocityregion2<Vr<3.5.TwodistinctvibrationregimesarerecognizedwithmaximumresponseforceamplitudesatVr=2.54andVr=10.16forgateopeningsa/D=0.48anda/D=0.86,respectively.Theresultsshowthatthe gate with perforated bottom profile significantly reduces cross‐flow vibrations in theregion 2 <Vr < 3.5. Although not exhaustively covered, themeasurements give reason to
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believe that the same holds for the higher region atVr > 8.5. It was found that dampingappeared to be higher for high hydraulic heads, but this did not affect the comparisonbetweenthetwogatetypes.The obtained data setwas used to evaluate and improve the performance of a numericalmodel. Time‐dependent FEM simulations on a moving grid were performed to solve theRANSequationsfortheflowandthegatedisplacementofthemass‐springsystem.Aninitialmodel validation step for the added mass in still water showed reasonable estimatescomparedtothemeasurementsandreproducedthewallproximityeffect.Subsequently,asaformofcalibration,thesensitivityoftheresponsefrequencyinflowingwatertovariationofthemass factorwasused to select an appropriate solid density and spring stiffnesswhilekeepingthenaturalfrequencyinairequaltoexperimentalvalues.Then,twocasesofstrongvibrations were simulated from the low and high Vr regions. For the high Vr case withresponse frequency 4.7 Hz, vibrations with growing amplitude were reproduced for therectangulargatewhile themodifiedgatewithaholeshoweddecreasingamplitudes in thesameconditions.Thismatchesresultsfromthephysicalexperiment.Even though the numerical model does not capture all features relevant for a completerepresentation of the excitation mechanisms, the results nevertheless give valuableinformationabouttheworkingofthenewgatedesign.Thedownwardjetsthroughtheslotsremoveflowinstabilitiesfromthebottomboundaryandsmoothenslocalpressuregradients.Inparticular,thesimulationsshowthatforthestandardrectangulargate,entrainmentfromthewake into the zonedirectlyunder the gate leads to a transferofmomentum from theflowtothegate.Thismovement‐inducedexcitationisabsentinthegatewithslots,becausetheleakageflowpreventsthisentrainment.
6.7 Summary
Two‐dimensionalnumericalsimulationswereperformedwiththeFiniteElementMethodtoassesslocalvelocitiesandpressuresforbothgatetypes.Amovingmeshcoveringbothsolidand fluid domain allowed free gate movement and two‐way fluid‐structure interactions.Modelling assumptions and observed numerical effects are discussed and quantified. Thesimulatedaddedmassinstillwaterisshowntobeclosetoexperimentalvalues.Thespringstiffnessandmass factorwerevaried to achieve similar response frequencies at the samedrynaturalfrequenciesasintheexperiment.Althoughitwasnotpossibletoreproducethevibrations dominated by impinging leading edge vortices (ILEV) at relatively low Vr, thesimulationsathighVr showedstrongvibrationswithmovement‐inducedexcitation (MIE).Forthelattercase,thesimulatedresponsereductionoftheventilatedgateagreeswiththeexperimental results. The numerical modelling results suggest that the leakage flowdiminishespressurefluctuationsclosetothetrailingedgeassociatedwithentrainmentfromthe wake into the recirculation zone directly under the gate that most likely cause thegrowingoscillationsoftheordinaryrectangulargate.
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6.8 Additionalsimulations:flowvelocityandpressureofcase1
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Figure6.13.Simulatedflowfieldsofonecycleforcase1,originalgate,fz=10.1Hz.
6.9 Additionalsimulations:analternativeventilationdesign
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Figure6.14.Simulatedflowfieldofalternativeslotdesignforcase2,fz=4.7Hz.
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Figure6.15.Simulatedflowfieldofalternativeslotdesignwithgateinfixedposition.Themaximumglobal flowvelocitymagnitudeof thisgategeometry forcase2 is3.22m/sand2.36m/sforthefixedgatecase.
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7 Data‐drivenoperation6
7.1 Introduction
The introduction of informatics in the field of hydrodynamic modelling has led to manyapplicationsofdata‐drivenmodelling(DDM)andmachinelearning,studiedinthecross‐fieldof‘hydroinformatics’.AnoverviewofDDMinriverbasinmanagementisgivenbySolomatineand Ostfeld (2008). Quoting the proceedings of the first international conference on thetopic,“Hydroinformatics,understoodastheapplicationofadvancedinformationtechnologytotheproblemsoftheaquaticenvironment,evolvedthroughthe1980s,receiveditspropername in 1989 andwas given a first published expression in 1991.”(Verwey et al., 1994).MorerecentexamplesarePengeletal.2012,Pyaytetal.2011a‐b,Krzhizhanovskayaetal.(2011) and Melnikova et al. (2011), who combine physics‐based and data‐driven modeltypes intohybridmodels inorder todevelop integral earlywarning systems fordetectingdike instabilities andother flood safety issues.What sparked thisdata‐drivenparadigm isnot only the steady increase in computational capacity and power, but also the growingavailabilityandsophisticationofsensors.Infact,itisnotthehardwarebutthedevelopmentofdataprocessingalgorithmsthatmaybeconsideredthebottleneckinmostendeavoursofputtingthecollecteddatatointelligentuse.Inthischapterweworkwiththepremisethatthebarriergatecontainscertainsub‐optimalproperties with respect to dynamic loads. Reasons why this not unlikely were given inSection 1.1. Moreover, it is assumed that the FIV characteristics of the gate are not fullyknown. The aim of this chapter is to introduce a data‐driven system for monitoring andpreventingflow‐inducedgatevibrationsbasedondatafromsensorsonthegates.Thatis,amodel is built through data‐intensive system identification and the model use is forpredictionandcontrol(seeSection1.2).Anotherwayofsayingessentiallythesameisthatmeasurementsareusedtofeedaself‐learningsystemthatfacilitates(automated)decisionsforgateoperation.Thetrendofincreasinglyusingremoteoperationofgatesunderlinestheneedtocheckthatnolong‐runningvibrationsoccur;anautomateddetectionsystemwouldcoverthisneed.The past decade has seen several research groups working on the design andimplementationofdecisionsupportsystemsforthecontroloffloodingrisks.TheEUprojectUrban Flood (2012) made meaningful progress by proposing an artificial intelligenceenvironment as a central component in an early‐warning system that checks the statesoffloodbarriersanddetectsanomalies.Afive‐yearprojectwithasimilarscope,FloodControl2015(2012)consistedofaconsortiumofDutchwater‐orientedcompanies,butdidnotlook
6Thischapterisbasedonandusestextandcontentfrom“Controllingflow‐inducedvibrationsoffloodbarriergateswithdata‐drivenandfinite‐elementmodelling”byC.D.Erdbrink,V.V.Krzhizhanovskaya,P.M.A. Sloot, published in: Klijn, F. & Schweckendiek, T. (Eds.) “Comprehensive Flood RiskManagement”,CRCPress/Balkema(Taylor&Francisgroup),Leiden.Proceedingsofthe2ndEuropeanConference on Flood Risk Management FLOODrisk2012. 20‐22 November 2012, Rotterdam, TheNetherlands,pg.425‐434.
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intothedynamicsofhydraulicstructureswithgates.TheLeadingScientistProgramme(LSP)comprises the development of an Advanced Computing Laboratory (2012), which hasdeveloped a computational infrastructure for studying complex systems and providingdecision support, among others for flood defences. A literature survey revealed only onecomparablestudyofaproposaltousesensorsforthemonitoringofgatedynamics,reportedby Han et al. (2011). This proposal does not include artificial intelligence or control,however.Firsta fewremarksaboutmeasurementsaremade inSection7.2.Then inSection7.3 theoverallset‐upofthesystemisgiven.Sections7.4reflectsontheapplicabilityofthesystemanddiscussespossibleextensions.Section7.5drawsconclusions.
7.2 Whatquantitiestomeasure?
Thefirststepistodecidewhichphysicalquantitiesshouldbemeasured.InasituationwherevibrationsareobservedoroperationaldifficultiesarisethatarepossiblytheresultofFIV,itis common practice to carry out field measurements. A good approach is to installaccelerationsensorsonthegatesandmonitorthegatebehaviourforacertainperiodoftimeduring which a representative range of conditions occurs. Figure 7.1 shows a temporaryinstallation of an accelerometer that was used during several sessions of in situmeasurementsspreadoutoverafewmonths.
Figure7.1.Temporaryinstallationsofaccelerationsensorsonthegateofahydraulicstructure.Bothtypesmeasureaccelerationinonedirection,asshownbytheblackarrows.PicturesbyDeltares.Fortheresponse,accelerationisgenerallyeasiertomeasurethandisplacementorvelocity.Foraharmonicoscillationthesethreequantitiesarenaturally linkedbydifferentiation(orintegration)with respect to time. In the complex notation, differentiatingwith respect totime corresponds tomultiplicationwith a factor iω. Support forces and pressures on thesubmerged gate are not easily measured in prototype, therefore the acceleration istakenasthemainstructuralresponsesignal.Moreover,thegateliftingheightorequivalentlythegateopening in timea(t)shouldbemeasured.Thisquantity isusuallypartof thegateliftingsystemanyway,soitjusthastoberegistered.
gatebody
accelerometerinwaterproofcase
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For the hydraulic conditions, it is necessary tomeasure upstream anddownstreamwaterlevelsh1(t)andh2(t).Thesequantitiesarerelativelyeasilymeasuredforlongperiodsoftime.Main point of attention is to choose suitable measurement locations, so as not to beinfluencedby the surface rollerorothereffectsdirectly related to the structure, includinglocalbathymetrychanges.Stationarymeasurementsofpointvelocitiesnearastructuredon’tmakemuchsense.Thiswouldhavetobedoneatagreatmanylocationsinordertogiveauseful picture of the flow field. There exist different techniques tomeasure the dischargethroughaconfinedcross‐section.Monitoringtheflowdischargepastagategivesveryusefulinformation, for one because it would allow a data‐driven model equivalent of the flowimpactmodel cascadediscussed inChapter4.However, accuratedischargemeasurementscanbecostly; someequipment is sensitive to sedimentationanddebris, andamulti‐gatedbarrierwouldneeddischargemeters in every gate.Moreover, onehas to be aware that asingleQ‐Hrelationdoesnotdescribeallstatesofanunsteadyflowduetohysteresiseffects(e.g.Travašetal.2012).Inthischapterdischargesignalsarenotused.The step from incidental prototype measurements to permanent gate monitoring is notexpected tobe complicatedhardware‐wise– itmostly relieson the installationofdurableaccelerationsensors.Theuseofwirelesssensorswouldbeagreatimprovement.
7.3 Systemset‐up
7.3.1 OverviewofthecomponentsThe proposed system consists of a control loopwith a numericalmodel fed by data fromsensorsattachedtothegate.Fornowasinglegatewithsubmergeddischargeisconsidered.Otherthanusingtheverticallymeasuredgateopeningasameasureforgateposition,thereis no principal reason to restrict the discussion to underflow gates. Figure 7.2 shows thefunctionalblockdiagramof the system.The shadedgrey rectanglemarks the componentsthatarediscussedinthischapter.
Figure7.2.Schematicoverviewofthecontrolsystemforgatedynamics.Thecomponentsinsidethegreyrectanglearediscussedinthischapter.Thevariableswithhatsarepredictions.
ydom,i
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sensors measure process
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The starting point is the gate itself, which holds a certain position that is assumed to bedescribed by its gate openinga. The “data acquisition”module collects data from sensorsinstalledonthegate, seeprevioussection.Theacquiredsignalsare input for the “primarydataanalysis”module.Itconsistsofappropriatesignalprocessingmethods(seeSection5.3)to derive the dominant frequency fdom,i and mean displacement amplitudes Ai from theaccelerationsignalsofthemostrecenttimewindow.Subscriptsirefertotheorientationandlocation of the sensor that produced the signal – for the one d.o.f. process discussed inprevious chapters this simply corresponds to the vertical direction. The hydraulic headdifference∆h=h1–h2andthemeangateopeningaduringthistimearealsoregistered.Thecentralmodule is the“machine learning”(ML)modulewhichconsistsofadatabaseofpriorstatesplusasetofartificialintelligencealgorithmsthatclassifyandinterpretdata.TheML module receives data from several sources. It receives input about the most recentsituation (achievedhydraulic conditionsandgate status) from the “primarydataanalysis”module.Furthermore it receives the latestpredictionsofwater levelsonbothsidesof thegate 1and 2fromaseparate,externalnumericalmodeloftheflowsystematalargerscalethatproduceswaterlevelpredictions.Thecomputedwaterlevelsofthissystemareusedtopropose the next gate opening anew needed to accomplish certain far‐field targets (e.g.minimum water depths for navigation). The ML module has the goal of determining theexpectedgatebehaviourof theproposedsettinganew fromthe recentlyachievedstateandthepredictedwater levels. If therecentlymeasuredsituationcloselyresemblespaststatescaptured by data already present in the database, it will be relatively easy to draw aconclusionaboutthestabilityofproposednewgatesettings.Ifnecessarythough,theMLunitwillsuggestanalternative,safergatesettingoropening/closingscenario.Thenthisinformationisusedinadecisionstepthatchoosesthemostsuitablescenarioandprescribes (designs) the requiredoperation.Thedecision step is shown inFigure7.2asaseparateblock,because itmaynotalwaysbeautomated.Thedesignedmeasure is carriedoutbytheoperationmechanism(controller)andthegatewillattainanewposition.Thenewsituationwillagainbemonitoredbythesensors,thuscompletingthecontrolloop.Thereisanoptionalextensiontothissystem:theMLmodulecanreceivecomplementarydatafromanothernumericalmodelthatperformsphysics‐basedsimulationsandproducesaresponseprediction.Thepurposeofthisistofillinregionsofmissingdata.The system adopts a straightforward way of vibration control: avoidance of criticalparameterrange.Thisisdonebyadjustingsettingsofgateopening,byselectionofstartingtimeof the operation, or by adjusting the speedof the gatemovement (this isnot alwayspossibletechnically).7.3.2 MachinelearningmoduleFromaMLperspective, the sensorydata (andpossibly the simulationmodel)provide thedatabasemodulewith labeleddatapoints.TheMLoperationsonthisdataaretherefore intherealmofsupervisedlearning,morespecifically,theyareclassificationoperations.The reduced flowvelocityVr helps toclassify flow‐inducedvibrations indifferentphysicalregimes (as seen in Chapter 5). Computing it as 2 ∆ is convenient and
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adheres to past studies. The acquired data (measured or calculated) occupies the three‐dimensionalspaceshowninFigure7.3(left).
Figure7.3.Leftplot:typicalresultsofvibrationsmeasurements.Vrisreducedvelocity,aisgateopening,Aiisgatedisplacementamplitudedetectedbysensor i.Rightplot:theproblemofpredictinggateresponsefor increasingVratconstantgateopening. In this figureand inFigure7.4, fictitiousdata is shown forillustrationpurposes.For different gate openings a1 and a2, the gate response is quantified by displacementamplitude as function of the Vr number. In this example, the strongest vibration at gateopeninga2isfoundataboutVr=2.6.Now,theproblemthatthecontrolsystemshouldtackleisshownintheotherplotofFigure7.3.Supposethegatehasaconstantposition,sothatthea‐dimensiondropsout,andtheheaddifferenceincreasessuchthatthedynamicstatemovesfromVr,1 toVr,2, thequestion is thenwhatresponse isexpected ifVrcontinuoustogrowtoVr,3.TheempiricaldatapointsoftherecentstatesbetweenVr,1andVr,2andtheknownstatesforVr>Vr,3maybewrong indicatorsof theunknownstateatVr3.Obviously, interpolationbetween the nearest known states in the (Vr, A)‐plane will not predict the state at Vr,3correctly.Toenable theuseofefficientartificial intelligence,we introduceacriticalmeanamplitudethreshold xcrit (see Figure 7.3), belowwhich the gate is said to be in a safe condition andabove which vibrations are harmful. This allows a transformation of the problem into abinaryclassificationproblem:alldatapointsbelongeithertoa‘critical’or‘unsafe’class(A>Acrit)ortoa‘safestate’class(A≤Acrit).Thisisinessenceaprojectionontothe(Vr,a)‐plane.Figure7.4showsthebinarydatapoints in thenewplane. In this form,closureoropeningunderconstantheaddifferenceisaverticalline.Achangingheaddifferenceintimeatfixedgate opening is shownas a horizontal line in this plane. The projected binarydata pointsconstituteatrainingset.Thetrainingobjectsarecharacterizedbytwoattributes(Vranda)andonelabel("safe"or"unsafe").Mostentrieswillbesafesituationsandincertainregionsislandsofcriticalvibrationstateswillbefound.
a
0
a2
a1
Ai
1 2 3 4 Vr Vr
Ai
Acrit,i
Vr,2 Vr,3
known stateunknown state
Vr,1
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Figure7.4.Fictitiousexampleofmeasureddatapoints inthe(Vr,a)‐plane.Vibrationsabovethecriticallimitareindicatedwithtrianglesandvibrationsbelowthecriticallimitwithsquares.Thisdataisusedastrainingdataforclassificationofnewstates(+).Aclassificationalgorithmisrequiredthatiscapableofmakingaccuratepredictionstowhichclassanewpointismostlikelytobelongto.Thereexistmanysuchalgorithms.Abasicchoiceis theK‐nearest neighbours (KNN) algorithm (e.g. Rogers andGirolami, 2012 andBishop,2006).Thisnon‐probabilisticclassificationmethodassociatesanewpointwithalabelbasedonthemajorityofthelabelvaluesoftheKdatapointsnearesttoit.Inthisalgorithmthereisin factno specific trainingstageother than the constructionofamatrixwithalldistancesbetween thepoints.ParameterKmustbe chosencarefully as itdetermines theamountofsmoothing; ifK is too lowthere isariskofoverfittingand if it is toohighunderfittingcanoccur.Here, it isproposed todetermine thevalueofK byaN‐fold cross‐validation, i.e. byrepeatedlyusingdifferentpartsofthetrainingsetasvalidationdata(alsocalledtestdata).Thedataset isdivided intoNsubsetsofequalsize.Onesubsetplaystheroleofvalidationdata and the otherN‐1 subsets are the training points. This is doneN times, so that allsubsetsplaytheroleofvalidationdataonce,producingoneerror(definedastheproportionof incorrect classifications) each time. Below a concise pseudo‐code for cross‐validationappliedtotheKNNalgorithmisgiven.
For K = 1 to k % loop for K values (using only odd values) For fold=1 to N % loop for N-fold cross-validation randomly sudivide data in folds D = [empty] construct matrix D of Euclidean distances of training data D = sort(D) For j=1 to K % loop of actual KNN find distances of j closest points check binary label value of closest points End determine majority of votes of closest points, this is the predicted label compute error of this fold End
Vr
a
0
new state
Ai < Acrit,iAi > Acrit,i
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compute error for this K as average of the N fold errors End
Atypicalquestionthesystemshouldprovideananswertoiswhetheritissafetoopenagateunder certain conditions. First,Vr should be determined usingmeasured∆h and responsefrequency f estimated from nearby available data or using the gate’s natural frequency.Figure 7.4 shows two options of opening scenarios that the system needs to evaluate,indicatedby+'s,thesearenewpointsthatrequireclassification.InthecaseofopeninginthelowerofthetwoVrconditions,thesystemmaygiveadvicetowaitwithgateoperationuntilthe expectedVr is reduced, in order to stay clear from the unsafe zone. Alternatively, thesystemmaydecidetoopenthegateatafasterrate,sothatthecriticalzoneisvisitedonlybriefly.7.3.3 Physics‐basedmodelThegoalofemployingaphysics‐basednumericalmodelinthiscontrolsystemwouldbetoprovidethedatabasewithtrainingdatapointsinareaswherenomeasureddataisavailable.Nexttowaterleveldataforboundaryconditions,pastresponsedatawouldalsobeavailabletothismodel,makingitessentiallyapredictionmodel.However,itwasfoundinChapter6thatsettingup,validatingandapplyingacomputationalmodelforsimulatingfluid‐structureinteractionrequiresalargeeffort.Itwasconcludedthatitdefinitelyhasvalueforanalysisofthe(excitation)process,butthatitnotnecessarilygivesasoundrepresentationofdampingandamplitudes.Therefore,employingaphysics‐basedmodelforproviding and asshown in Figure 7.2will be challenging. The best approachwould be to carefully select alimitednumberofunknownstatesandattempttousetheanalysisbasedontheoutcomesofthisphysics‐basedmodeltoevaluatethegatestability.Thefactthatrunningandinterpretingthismodelistime‐consumingandhardtoautomate,makesitasyetunfeasibletoincludeitintheoperationalcontrolchain.
7.4 Resultsofexperimentaldataclassification
TheprocedureofSection7.3.2isappliedtotheexperimentaldatafromChapter5.Thedatasetusedherefortheclassificationconsistsof145observedvibrations.Figure7.5showsthisdatasetinthesamewayasdepictedbyFigure3.Thedimensionlessdisplacementamplitudewasfoundby ⁄ ⁄ ⁄ .Thedatarepresentsmeasurementsatvariousgateopenings.Duringeachtest(correspondingtoonedatapointinFigures7.5and7.6),thedischarge,gateopening and hencewater levelswere kept constant – the same can be assumed for fieldsensordataovershorttimewindows.
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Figure7.5.ExperimentalvibrationdataoftheoriginalgatetypefromthephysicalscalemodelofChapter5.Theplotshowsnon‐dimensionaldisplacementamplitudeasa functionofreducedvelocity forsteady‐state conditions. The data set is divided into two classes using a threshold at A/D = 0.004. One highamplitudeoutlieratVr=10isnotplotted.AsknownfromChapter5,therearetworegionsofVratwhichthevibrationsgrowinsizeconsiderably:2<Vr<3.5andVr>7.A thresholdatA=0.004Ddivides thedata into twoclasses named "vibrations" and "no vibrations". The signals below this threshold containirregular vibrations of small amplitude, above itmore regular vibrations are found. Usingthis binary division, the experimental data is plotted as functions ofVr anddimensionlessgateopeninga/DinFigure7.6(left).
Figure 7.6. Left: binary identification of vibration regions: reduced velocity versus dimensionless gateopening.Right:resultofcross‐validationofclassificationshowingthatK=3givesthebestclassification.ItcanbeseenfromFigure7.6(left)thatvibrationsareactuallyfoundforspecificregionsofVr incombinationwithcertaingateopeningranges.Whileobviouslythe limitedamountof
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datadoesnotgivethefullpicture,itisforexampleshownthatvibrationsintheregion2<Vr<3.5vanishwhenthegateisopenedto1.4timesthegatethicknessandhigher.ThesamecannotbeconcludedfromtheavailabledataforvibrationsdetectedatVr>7.Toclassifythisdata,itwasfirstnominalizedtoscalesfrom0to1.Aten‐foldcross‐validationwasthendoneonthevalueofKaccordingtothealgorithmofSection7.3.2.ThisshowedthatK=3isthechoicethatgivestheminimalerrorinthiscase,asplottedinFigure7.6(right).ThecodethatwasusedisanadaptationfromMatlabscriptsbyRogersandGirolami(2012).ThisKNNalgorithmwithK=3 isnowready tobeusedasanautomated tool for trackingcriticalgateconditions.Thecontoursofvibrationregionsarerepresentedbyclassificationdecisionboundaries,uptoacertainaccuracy.Thepurposeofthistechniquehasoverlapwiththat of physical model tests, but it automates and hence accelerates the identification ofsignificantdynamicresponseregions.
7.5 Applicationchallenges
7.5.1 GeneralSomeremarksontheproposedsystem:‐ThesystemoutlinesofarassumedauniqueresponseforonegateopeningandVrvalue.Inotherwords,theclassificationinSections7.3and7.4wasbasedontheassumptionthatthefunctionf:(a/D,Vr)→A/Diswell‐defined.ButmanydimensionlessparametersfromSection2.4.2 are missing here. A more easily measured alternative to Fr is the submergence CsdefinedinSection5.5.Includingthisparameterasaclassificationlabelmeansincludingtheeffect of the varying free surface. Also, the amplitude is intimately linked to damping.IncludingScorζdoesnotmakesense,however.Itwillbepracticallyimpossibletomonitortheseinprototypeandbesides,asinglestructurewillexperienceverysimilarhydrodynamicdampingforonehydraulicconditioneachtime.Smallchangesintimeofstructuraldampingwillbecapturedautomaticallybychangesintheself‐learningmodel.Thesameholdsforallstructural parameters that gradually change with time as the structure is aging. Inconclusion, theMLmoduleworksbest if themappingA/D= f (a/D,Vr,Cs) is indeedwell‐defined. So we end up making a classification in three‐dimensional space, which iscomputationallyquite thesameaswhatwasdone inSection7.4.Theexperimentaldata isnotsuitablefortesting3DclassificationbecauseitcontainsverylittleCsvariation.‐ A challenge for the classification algorithm is that the training set is very asymmetric,becausetherewillalwaysbemanymoredatapoints forsafesituations than forsituationswith regular high‐level vibrations. The experimental data set is in this respect notrepresentative of real‐life monitoring data, as the experiment focused on finding andrecordingvibrationregions.‐Asecondaspectthatisratherlikelytobedifferentinprototype,istheoccurrenceofmorethanonevibrationregion.Thereweretworegions(fixedbyvaluesofreducedvelocityandgate opening) in which significant vibrations are found in the experiment, by activelychangingstiffness.Thisdoesnotmeanthatoneregioncannothavemultiplephysicalcauses(excitationmechanisms)simultaneously.
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‐ The system is built with ‘onlineML’, a self‐learning process that gradually improves bytakingupnewsensordataastimeprogresses.Therateatwhichthedatabaseisfilledwithdatapointsofcriticalsituationsdependsonchosencriticallimits(threshold)andoccurrenceofextremeevents.Collectedvibrationdataofrealbarriergateswillatfirstnotbedistributedinawaythatmakesdeterminingtheregimeboundariesstraightforward.Inparticular,waterlevel variations and gate openings are usually not sufficiently varied during everydayoperation to get a complete view on vibration states. Dedicated measurement events inwhich gate openings are prescribedwhen specific hydraulic heads occur are useful, if notnecessary,tocollectspecificdatatofillinblankspots.‐ Every time a gate condition is visited thatwas alreadymeasuredbefore, thedatabase isupdated. If thenewlyvisitedconditionwasanunknownpointpreviouslysimulatedbythephysics‐basedmodel,thenewmeasurementsalsoactascalibrationdataforthismodel.‐Thestrategyparametersoftheclassificationoperationarelinkedwithasafetyfactorthatfollowsfromthequestionofwhattheconsequencesareofmakinganincorrectclassification.Optimisation of the threshold value of the binary split and the parameters of theclassificationalgorithm(fortheKNN‐algorithmthisisK)shouldtakethisintoaccount.7.5.2 MultipledegreesoffreedomThe application of one sensor on a gate matches the single d.o.f. (SDOF) schematization,providedthevibrationmodeisknown.Themostcommongatevibrationmodesareverticaltranslation, horizontal translation and rotationalmotion (for tainter valves),which are alldescribed aptly by the single mass‐spring analogy. This was discussed in Section 2.5.Ultimately, the hydro‐elastic conditions determine if a certain vibration mode actuallyinitiates.Anyprototypeinvestigationshouldfirstfocusonidentifyingtheprimaryvibrationmode.Theuseofsensorscanfacilitatethisprocessandconfirmanalyticalstudies.Gateswithconsiderable longitudinal spans can suffer from bending vibration modes (Ishii, 1992),which are found only when additional accelerometers are placed strategically and theirsignalsareanalysedjointly.ItisremarkedbyKolkmanandJongeling(1996)thattherisingdemand for structures with larger dimensions results in relatively weaker, i.e. less stiffstructuralelements,thusincreasingtheriskforvibrations.There could be more than one significant mode of vibration. Systems of multiple d.o.f.(MDOF) have n > 1 distinct eigenmodes and eigenfrequencies (for n d.o.f.s) and areanalyticallydescribedbyamatrix‐versionofthemotionequation:
, (7.1)
wherethecapitalsarenxn‐matricesofmass,dampingandstiffness,respectively,w(t)istheresponsevectorandf(t)istheforcingvector.TheaddedmassmatrixalreadymentionedinSection2.4hastobeincludedinM.Anarbitrarydampingmatrixwillcausedisplacementsinoned.o.f.influencingtheforcesonanother:coupling.Thesolutionof(7.1)hastheform
∙ , (7.2)
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with ei the eigenvectors and αi(t) ‘participation functions’ giving the contributions of theforcevectortoeachd.o.f.(KolkmanandJongeling,1996).TherelativeeaseofanalyticalgeneralisationfromtheSDOFmechanicalvibrations(Section2.3)tomultiplemodesofvibrationconcealsthefactthatanalyticaldescriptionsoffluid‐solidinteractions(Section2.4) formultiplecoupledmodesarealwayscomplexandthephysicalunderstanding of MDOF FIV systems is incomplete. The study of gate vibrations ofsimultaneous cross‐flow plus in‐flowmodes by Billeter and Staubli (2000) illustrates theextracomplexityofexperimentation.Moreover,theyclaimthattheadditionofonlyoned.o.f.increased the region of conditions in which non‐linear response can be expected. Thisimplies thathopes for tackling forwardMDOFproblemsareslim;physics‐basednumericalmodellingforreal‐lifeMDOFconditionsispracticallyoutofreach.Thisisanotherargumentinfavourofthedata‐drivenapproach.Letussupposeit isnotclearwhatthemoststringentmodesareandplentyofsensorsareinstalled inanattempttocatchthese.Thenfrequencydomaindecomposition(FDD) isonewayof findingthemodesfromthecollectivematrixofallmeasuredsignals.Brinckeretal.(2001)showshowtoapplythisforamodalidentificationusingonlyoutputsignals.Thisisarelativelyeasycomputationthatdistinguishesseparatevibrationmodesfromasetofsignalsandcanbeaddedtothe‘primarydataanalysis’block.7.5.3 ApplyingmachinelearninginengineeringEventhoughcomputationalhydraulicsresearchhasmadeuseofartificial intelligencefromrelatively early dates (Verwey et al., 1994), state‐of‐the‐art ML applications are notcommonplace for real‐life (consultancy) projects in the conservative field of hydraulicengineering.Approaches thatbypassprocessanalysisandexpert judgementareoftenmetwith scepticism.Beforea control systemsuchas theonepresented in this chapterwillbetrusted to replacemanual operation, it has to be thoroughly tested in pilot projects. Seenfromtheoppositeside,MLissadlyabusedfrequentlyasaquickwayofproducingscientificpapers – thishas resulted in anuncontrollable growthof superfluousML ‘tricks’ thatwillneverbepickedupbyothers, letaloneseeany formofapplication.Attempting to reversethis trend, Wagstaff (2012) presents two necessary conditions for producing “ML thatmatters”: (i) insteadof testingnewalgorithmson isolatedbenchmarks,choosemeaningfulandrelevantdomaintests;and(ii)communicatetheresultsbacktotheproblem’sresearchfield.Anumericalmodel inML is trainedonadatasetby consideringerrorsbetweenpredictedandactualdatapoints.Topreventthemodelfrombecominga‘one‐trickpony’thatperformspooronotherdataofasimilarproblem, it iscustomtosplitthedataset intoatrainingsetandatest(orvalidation)set.Thetrainedmodelisthenappliedtothetestdatasettoproduceatesterror(orvalidationerror),whichisabettermeasureforhowthemodelwillactuallyperformonnewproblems (in the samedomain).However, asWagstaff (2012)pointsout,the fact thatacceptable levelsof testerrorsgreatlyvaryoverapplicationdomains isoftenoverlooked.
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7.6 Conclusions
This chapter has outlined and partly tested a data‐driven approach to vibrationcharacterisationforimplementationinanoperationalsystemthatcontrolsthegatepositionsuch that dynamic excitations are avoided. Gate response data from sensors undergo aclassification with the goal of predicting whether a future gate state is safe or not. ThepresentedMLapplicationfulfillsWagstaff’s(2012)conditionsandincludedcross‐validation.ThetestresultsuggeststhatintheKNNalgorithmonlyasmallnumberofneighboursshouldbeusedtodeterminetheassignedclass.Thesestraightforwardcomputationsreflectaproofof concept rather than an optimised algorithm choice. There aremany refined alternativeclassificationmethodsavailable.Theproposedbinaryclassificationofthevibrationregionsinthe(Vr,a)‐planeor(Vr,a,Cs)‐planeisanautomatedwaytorecognisetransitioncharacteristicsbetweenstableanunstabledynamicbehaviour.Itisreasonabletoassumethatthecontoursbetweensafe/unsafeclassescorrespond to transitions from irregular to regularoscillations (ona signal level)and toatransition from net energy dissipation to net energy transfer from the flow to the gatemotion (on a physics level). Ideally, the sensors are sensitive enough to record thesetransitions and thus produce training points of the unsafe statewithout actually enteringstatesthatarepotentiallydangerousforthestructure.Despitethefactthatphysics‐basedsimulationsleadtobetterinsightsintothegateresponsein unmeasured future conditions than the status quo of complete uncertainty facedotherwise, the incorporation of these simulations in an operational control loop seemsinappropriate.Theyaretootime‐consumingandrequirecareful interpretationthat ishardtocaptureinanalgorithm.Theirvaluelies inanalysisandpossibly inofflinegenerationofcomplementaryinputdatafortheclassificationdatabase.Finally,thissystemcanalsobeusedforstand‐alonefieldmonitoring,namelywhenthewaterlevel prediction, decision and operation blocks are disabled. As such it is a useful aid inanalysesbytrackingtheconditionsunderwhichvibrationsoccur.Thisknowledgeisusefulforimprovingaflawedgatedesign.
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8 Evolutionarycomputingforsystem identification7
8.1 Introduction
Classesofcomputationalmethodsforhydraulicstructuredynamicscanbecategorisedinasimilarwayas forgeneralstructuralanalysis,namelybydistinguishingdirectand indirectmethods inthe timeandfrequencydomain.The forwardapproachofChapter6 isadirectmethodinthetimedomainandhastheadvantageofcapturingpotentialnon‐linearities,butthedisadvantagethatonlyasingleconfiguration(definedbymassandstiffnessofthemass‐springandbythehydraulicboundaryconditions)canbeconsideredpersimulation.Thisishighly inconvenient if the added coefficients and response frequency have a mutualdependence – which was rightly assumed not to be the case in Chapter 6 because of anegligiblefree‐surfaceinfluence.FrequencydomainmethodshavebeenmentionedbrieflyinSections 2.3 and 7.5.2. The straightforward extension to more degrees of freedom is anadvantageofmodalanalyses.This involvestheassumptionofequaldampingdistributions,however,andnon‐linearitiesarenotapparentfromfrequencyspectra.Anapproachnotyetmentionedistheimpulseresponsemethod,whichderivestheresponsetoanarbitrarytime‐varyingforcefromtheresponsetoaunitimpulseusingDuhamel’sconvolutionintegral(e.g.Maymon,1998).Thiscomputationincludesthewholefrequencyrange,buttheobviouscatchisthattheunitimpulseresponsehastobeknownfirst.Soestablishedcomputationalmethodsforgatevibrationsmeetwiththreeobstacles:‐ The assumption of low positive damping, this is convenient computationally, butirrealisticinmanyscenarios.
‐ The possibility of mutual dependency between added coefficients and responsefrequency.
‐ The existence of non‐linear behaviour, caused for instance by the stiffness forcevarying non‐linearly with displacement (e.g. when the suspension consists of ahydraulic cylinder) or the combined time‐dependentworking of negative hydraulicdampingandincreasedstructuraldampingathighamplitudes.
Afourthpracticalcomplicationisthatoftenonlythegateresponsecanbemeasured,aswasthe case in theexperimentofChapter5, andnot the flow forceson thegate.This chaptertherefore looks into a newway ofmodel buildingwith the gatemotion represented as adynamicalsystem.Systemidentificationstudiesembarkedonthetaskof inferringODEmodelsa fewdecadesago (Åström and Eykhoff, 1971). However, the fixed structure acting as a vehicle for theparameteroptimizationwasusuallynotanODEitself,butratheraneasy‐to‐computebasisfunction like a polynomial. Non‐linear system identification techniques for structural7Thischapterispartlybasedontextsandcontentfromthepaper“Identifyingself‐excitedvibrationswithevolutionarycomputing”,ProcediaComputerScience,Vol.29,pp.637‐647,2014.ISSN1877‐0509.ThiswillbepresentedattheICCS2014conferenceinJune2014.
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dynamics are extensively described in Kerschen et al. (2006). There is no mention ofheuristic techniques; the evolutionary approach of this chapter does not appear in thisoverview.Theadventofmodernheuristicsandthesteadyincreaseincomputingpowerhasenormously boosted possibilities for regression of all kinds (e.g. by artificial neuralnetworks),butmanytechniquesdonotprovideclearinsightsintotheworkingofthesystem.In applied hydrodynamics, working with pitch‐black models is seen as an inconvenientdrawback.There isalwaysadesire tobuild feelingandconfidencealongwithbuilding themodel.The second‐order ODE of motion together with initial conditions form an initial valueproblem (or Cauchy problem). Solving it numerically, for example with Runge‐Kuttaschemes, isa forwardproblem: the timeseriesy t isunknown. If,on thecontrary,y t isknown,thenthe inversetaskof findingtheODEthatproducedit isasystemidentificationproblem,seeFigure8.1.
Figure8.1.TheproblemoffindingtheODEresponsibleforproducingagivenone‐dimensionaltimeseries.The ODE can have an unknown algebraic structure and/or unknown coefficients. This scheme isreminiscentoftheoptimizationschemeinSection1.2.If only the numerical coefficients of the ODE are unknown, but the structure of theexpressionisknown,anoptimizationproblemincontinuousspace(ofdimensionnequaltothe number of coefficients) needs to be solved. Let us look at a few instances of ODEs inFigure8.2.ODE 0 1 0 0
timedom
ain
known
y (t)initial conditions
ODE solver algorithm
(y0, y0)known
known
f( t, y, y, y ) = 0unknown
ODE
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phaseplane
Figure8.2.Vibrations in timedomain (upperrow)andphaseplane (lowerrow).Leftcolumn:constantnegativedamping;middlecolumn:non‐lineardampingbyVanderPoloscillator;rightcolumn:non‐linearmass term.The correspondingODE equationsarewrittenon top; y is the verticaldisplacementof themass.Theinitialstatesareindicatedbythickdots.ThefirsttwocolumnsinFigure8.2showquintessentialself‐excitation:anegativedampingconstantandtheVanderPoloscillator.Thelatterfamousexamplehasanon‐lineardampingtermandforhighenoughvaluesoftheparameter, so‐called ‘relaxationvibrations’occurwhichshowsuddentransitionswithshortmomentsofhighvelocityatcertainpartsof theperiod.ThethirdexampleintherightcolumnofFigure8.2showsanundampedoscillationwith a non‐linearmass term. A standardway of depicting non‐linearities is in the phase‐plane;adeformedlimitcycleisagoodtelltaleofnon‐linearbehaviour.The aim in this chapter is to explore evolutionary computing EC for uncovering ODEsdescribing non‐linear, in particular self‐excited vibration types. This identificationovercomesthefirstandthethirdobstaclementionedinthissectionandcertainlymeetsthecriterion of fostering ‘Fingerspitzengefühl’. In fact, knowing the exact ODE reveals veryuseful information for the analyst not found from Fourier analyses: non‐linear termsdependencies on frequency and added mass terms. The desire is to contribute to thedevelopment of a tool for detecting self‐excited vibrations before they grow beyond safelimitsandcausedangerouslyhighdynamicforcesonthestructure.Firstaridiculouslyshort introductiontoheuristicsandevolutionarycomputing isgiven inSection 8.2. Then Section 8.3 zooms in on differential evolution, which is applied to theproblemof system identification (Section 8.4). Section8.5 shares thoughts on and gives aproof of concept of the application of the presented method to the experimental data ofChapter5.Section8.6thengivesremarksongeneticprogrammingandsymbolicregressionandSection8.7drawsconclusionsandgivesanoutlookonfuturework.
8.2 Meta‐heuristicsandevolutionarycomputing
In optimization there are two kinds ofmethods: exactmethods that guarantee to find anoptimalsolution,andheuristicmethodsthatdonotgivethisguarantee.Theissuewithexactmethods is that they become practically unusable for larger and harder optimizationproblems.Heuristicoptimizationmethodsweredevelopedasfasteralternatives,butwithoutthe assurance of returning the optimal solution. Rothlauf (2011) remarks that standard
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heuristics“areproblem‐specificandexploitknownrulesofthumb,tricksorsimplificationstoobtainahigh‐quality solution”.Thesameauthordistinguishes threeclassesofheuristicoptimizationmethods:‐heuristics(dividedinconstructionheuristicsandimprovementheuristics)‐approximationalgorithms(whichprovideaqualityboundforthesolution)‐modernheuristics,alsocalledmeta‐heuristics.
A typical featureofmeta‐heuristics is the iterative improvementof solutions,while at thesametimesolutionsoflowerqualityareallowedtobepartofthesearch.Thishastodowithexploitation and exploration phases in the search. Meta‐heuristics are seen as general‐purposetechniquesthatcanbeappliedtomanydifferentproblems.A popular meta‐heuristic, evolutionary computing (EC) is a form of computationalintelligence inspired by nature. Evolutionary algorithms (EAs) are stochastic population‐basedsearchalgorithmswithacommonbasicloop:thereproductioncycle.AcomprehensiveintroductiontoEC isgivenbyEibenandSmith 2007 . Inshort,apopulationofcandidatesolutions (individuals) gradually evolves under the influence of one or more objectivesdictated by a fitness function. In the reproduction cycle, successively parents are selectedfromthepopulationtoengageinvariationoperations(recombinationandmutation),givingoffspring, from which eventually individuals are selected to move to the next generation(survivorselection).Figure8.3depictsa(debatable)familytreeofevolutionaryalgorithms.
Figure8.3.Familytreeofevolutionaryalgorithms,listedchronologicallyfromlefttoright.
8.3 Differentialevolution
Not generally considered as one of the classic EAs, differential evolution (DE) is acompetitive derivative‐free global optimization method introduced by Storn and Price(1997).Itsdistinguishingfeatureistheuseofdifferencevectorsinthemutationoperation.The global scheme of DE is shown in Figure 8.4, adopting the terminology by Das andSugantham (2011).This is themostbasic versionwith amutationbasedon threevectorsandstrategyparametersFandCR.
evolutionary algorithms
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Figure8.4.Globalschemeofbasicdifferentialevolution:definitionofvectors.IthasanaturalrobustnessthatmakesitstandoutfromsomeearlierEAs. Itsperformancehasgrownbyseveralimprovements(DasandSuganthan,2011),mostnotablybytheuseofparameter control. In this technique the strategy parameters change during the run, seeEiben et al. (1999). We apply a recent version of DE by Choi et al. (2013) that has self‐adaptive parameter control. They call it ‘Cauchy DE’ because the strategy parameters arevaried by drawing from Cauchy distributions. The pseudo‐code of the algorithm is givenbelow.
Initialization -Initialize population of NP vector individuals X1,G, … , XNP,G where Xi,G = [x1,i,G, x2,i,G, … , xC,i,G] and where C is the number of coefficients that are being evolved, G the generation (G = 0, ... , Gmax). Entries xj,i,0 are uniformly random from [-1,1] for i = 1, … , NP and j = 1, … , C. -Initialize control parameters CRi,0 = 0.25, Fi,0 = 0.6 (acc. to Choi et al. 2013) and adaptation parameters CRavg,0 = CRi,0 and Favg,0 = Fi,0 -Generate target data y(t), y’(t) and divide into training and test sets. FOR R = 1 to Rmax DO % run loop FOR G = 1 to Gmax DO % generation loop FOR i = 1 to NP DO % individuals loop Main loop: Differential Evolution -Determine fitness f(Xi,G) of individuals (see routine) -Mutation: generate a mutant vector Vi,G = Xr1 + Fi,G*(Xr2 – Xr3) from three donor vectors Xr1, Xr2,
Xr3 randomly selected from the individuals of generation G-1. -Crossover: generate a trial vector Ui,G composed of uj,i,G (j = 1, … , C) by applying the rule IF
rand[0,1] ≤ CRi,G-1 OR j = jrand THEN uj,i,G = vj,i,G , ELSE uj,i,G = xj,i,G-1 , where jrand is a random integer 1 ≤ jrand ≤ C and vj,i,G is an entry of Vi,G.
-Selection: determine fitness f(Ui,G) of trial vectors. IF f(Ui,G) ≥ f(Xi,G-1) THEN Xi,G = Ui,G and inherit associated fitness and control parameters, ELSE Xi,G = Xi,G-1 and leave fitness and control parameters unchanged.
END FOR -Update control parameters Fi,G and CRi,G by adding a randomly drawn number from the Cauchy
distribution Ca(0,0.1) to the mean value of the control parameters of all successfully evolved vectors. Truncate if necessary.
-Replace a non-fittest individual by a newly generated individual. END FOR
donor vector
initialization of parameter vectors
mutation
crossover
selection
target vector (parent)
3 random parameter vectors &
&
F
donor vector & CR trial
vector
fittest vector survives
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Post analysis Compute test error of run R by solving the ODE with the winning set of coefficients and determining
the mean absolute error of all predicted values compared to the test values. END FOR Compute mean duration and mean and min of test errors of all runs R…Rmax. Fitness computation -Insert the coefficients of each candidate vector in the fixed, assumed ODE equation structure. -Apply Runge-Kutta, with adaptive step-size and predetermined relative error tolerance for numerical
integration. -Fitness := -1*MAE*penalty, where MAE is the mean absolute error of the training data compared to
the result from solving the ODE with candidate coefficients. The penalty punishes candidate models for which the integration failed to determine values at all training times, penalty := ((size of training set – size of candidate set) / size of candidate set)*100 and penalty = 1 if the integration was completely successful.
8.4 Identifyingself‐excitedvibrationsusingdifferentialevolution
8.4.1 ApproachTheaimofthissectionistoidentifyvibrationsfromonlyadisplacement(output)signaly(t).Thatis,withoutusingsignalsofpressuresonthegates(input).Thisispreferablydoneinaway that permits speed‐up to practical time frames for early‐warning systems. It will beassumedthatthemainpartofthestructureisalreadyknown:thesecondorderODEforallvibrations without external forcing, with optional unknown non‐linear terms. DifferentialevolutionisusedtooptimiseasetofcoefficientsoftheseODEs.Ageneratedsyntheticdatasetisrandomlydividedintoatrainingsetandatestset,basedonachosenpercentageofdatatobeusedfortraining.Aftertheevolutionhasended,theunseentargetpointsareusedtoquantifythepredictivepowerofthecandidatemodelbycomputingatesterror. Inordertocomparethetesterrorsofdifferenttargetdatasets,wenormalizethemeanabsolutetesterror(MAE)asfollows:
, ≔1 1 1
| |,
, (8.1)
where containsthepredictionsand thetargetvaluesfortesting, isthemeanof andisthestandarddeviationof .Allresultsinthefiguresandinthetableofthispaperare
given as normalized mean absolute test errors (NMAE). The computations were doneunparallellizedonasingleInteli7processor,2.93GHz,8GbRAM.The numerical experiments consist of three parts: a validation case, the self‐excited casesandasensitivityanalysis.TheresultsarereportedinSections8.4.2and8.4.3.The case of forced vibrations for a linear system with constant coefficients is used forvalidation:
sin , (8.2)
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wherey isthedisplacement,t istime(theindependentvariable)andallothersymbolsarephysicalconstants.Newtoniannotationisusedfortimederivatives.Togetherwiththereal‐valuedinitialconditions and ,equation8.2constitutesaninitialvalueproblemthatwillbesolvedintwoways:(i)non‐linearregressionontheanalyticalsolutionand(ii)regressiononafixedODEstructure:The first approachuses the sumof the general andparticular solutionof forcedvibrationwithviscousdampingasanassumedequationstructure:
sin sin , ∈ (8.3)
ThesecondapproachstaysatthelevelofODE:
sin , , , ∈ (8.4)
For both approaches, the coefficients are initialized randomlybetween ‐1 and1. They arestoredinaseven‐dimensionalvectorandoptimizedviaDE,asdescribedinthepseudo‐codeinSection8.3.Avariationofthesecondapproachwhereonlyfivecoefficientsareevolvedisalsoconsidered,wheretheinitialconditions(IC)areassumedknown.Practically all non‐linear vibration problems defy full analytical treatment, so there is noclosed‐formequationavailablefory(t).FortheseproblemsweworkwiththeODEstructure
0, where massm or damping c are replaced by a first or second orderpolynomialterminytoaccountforthenon‐linearity.TheresultsofthisaresummarisedinSection 8.4.2. Also, a non‐linear mass system and a system with time‐varying stiffness(Mathieuequation)areexamined.Theseareallunforcedoscillatorswherechaosdoesnotplayarole.8.4.2 ResultsValidation:ForcedvibrationswithconstantcoefficientsThefollowingtargetfunctionisdefinedforthevalidationruns:
2.5 0.35 0.3 3.0sin 2.2 , with initial conditions 0, 0.25. (8.5)
Figure8.5(left)showsthesampledtargetdata‐setandanexampleofacandidatesolution.Thetotaldatasetconsistsof465points,meetstheNyquistcriterion,andissplitintrainingandtestdataindifferentratios(inFigure8.5halfofthedataaretrainingpoints,andhalfaretestpoints).Populationsizewassetat80and250generationswerecomputed.
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Figure8.5.Differentialevolutionappliedtoregressiononthesignalofaforcedvibration.Left:thetargetdata.Inthisexample50%ofthedataisusedfortraining.Right:Thebest‐so‐farfitnessofthreeruns,foranassumedstructureoftheanalyticalsolutionandfortheODEstructure.The right plot in Figure 8.5 shows three examples of how the solutions improved withgenerations.Thetwoappliedexpressionstructureswerelaidoutintheprevioussection,theonlynecessaryadditionisthatthefitnessevaluationoftheanalyticalsolutionstructurerunsdiffers from thepseudo‐code inSection8.3because there isnoneed to solveanODE; thecandidate values follow right away after substitution in the assumed y(t) expression. Theresultingcoefficients reflect themultimodality, since forexamplesin(t)=sin(t+2). Itwasgenerally found that the less successful computed functions capture the low‐frequencydampedfreevibrationquitewell,butgivearatherpoorestimateoftheforcedvibration.Figure8.6belowgivesanoverviewof the resultsbasedon10runsperplottedpoint.TheplotontheleftgivestesterrorsexpressedasNMAE,accordingtoequation8.1.Theplotontherightshowstheaverageruntimeinseconds.
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Figure 8.6. Results of evolving the forced vibration based on analytical solution structure and ODEstructure as a function of the percentage of data used for training. Left: test errors (NMAE); right:computing time. IC stands for initialconditions,P stands forpopulation size.Everypointrepresentsanaverageof10runs,foranevolutionof250generationswithapopulationof80;exceptfortheODEwithinitialconditions,forwhichonlyfivelengthyrunsweremadeforeachtrainingdataset.The plots show that the analytical structure requires the least computation time, but it issignificantly less accurate than theODE structurewith five evolved coefficientswhere theinitial conditionsareknown (“ODEwithout IC”).Theanalytical structure ismore accuratethantheODEcasethatalsoevolvesthetwoinitialconditions(“ODEwithIC”).Anattempttoreduce the computation time for the ODE structure by using a smaller population of 40(“ODEwithoutIC,P=40”)resultedinhighertesterrorsandcomputationtimescomparabletotheanalyticalruns.Theresultsshowthatincludingorexcludingthetwoinitialconditionsmakesnodifference forcomputationtime.Additionally, thevalidationproves that there islittleoveralldependenceonthepercentageofdatausedfortraining.Thetesterrorsareonlyslightlyworsewhen less than 40%of the data is used for training. Computational factorsrelatedtotheconvergenceoftheDEalgorithmandtheODEsolutionprocessareapparentlydominant.Inparticular,itwasfoundthatthesettingsofrelativetolerancethatdeterminethenumber of iterations of the ODE solver during the error computation have a profoundinfluenceonthestandarddeviationoftheachievedtotalruntimes.MainrunsTable8.1showstheresultsofcomputingcoefficientsforvariousself‐excitedvibrations.Allcomputationshadapopulationof50individualswith120generationscomputed,andused50%of500datapointsfortraining,andtheremainderfortesting.ForeachcasethemeanandstandarddeviationoftheNMAEtesterroriscomputedover25evolutionaryruns.
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Table8.1.Computationresultsbasedonnormalizedmeanabsoluteerror(NMAE).#Cdenotesthenumberofevolvedcoefficients.Foreachcase,25runsof120generationsweredonewithapopulationsizeof50,using250trainingpointsand250testpoints.
vibration targetODE #CaverageofNMAE
standarddeviationof
NMAE
linear,constantcoeff.* 2.5 0.35 0.3 3.0 sin 2.2 5 39.2∙10‐3 69.5∙10‐3
negativedamping 0.1434 0 1 3.44∙10‐3 0.89∙10‐18 2 3.07∙10‐3 1.77∙10‐18 3** 3.23∙10‐3 1.88∙10‐3
non‐lineardamping 0.450 2.728 1.903 0 3 287∙10‐3 373∙10‐3
VanderPoloscillator 1.2218 1 0 2 3.13∙10‐3 0.42∙10‐3 3 3.26∙10‐3 0.82∙10‐3
non‐linearmass 1.7120 1.4815 0.4903 0 3 3.28∙10‐3 0.0776∙10‐3
Mathieuequation*** 0.25 0.34sin 2.18 0 3 70.8∙10‐3 25.9∙10‐3
*BasedontheODE‐runswith50%trainingdataandwithoutevolvinginitialconditions.**OnlythreerunsweremadeduetopoorconvergenceofODEsolver.***TheMathieuequationdescribesnotself‐excitedbutparametricallyexcitedvibrations.
TheresultsinTable8.1showthattheconstantnegativedampingandnon‐linearmasscaseshave low test errors compared to the validation case of the forced linear vibration withconstantcoefficients.Moreover,theirtesterrorsshowverylittlevariation.Forthenegativedampingcase,itmakesnodifferencewhethertheconstantcoefficientisfoundusingasinglecoefficient,C1,ora linear termwith twocoefficients,C1+C2y,orasecond‐orderpolynomialtermwiththreecoefficients,C1+C2y+C3y2.Similarly,theVanderPoloscillatorshowsasmall,insignificant deteriorationwhen a linear term is added to theC1+C2y2 term that is strictlyrequired.Thepoorresultforthenon‐lineardampingcaseisduetosuboptimalconvergenceofeightrunsoutof25.ExtendingtherunstomoregenerationswillmostlikelyimprovethemeanNMAE.The samecanbe saidof theMathieuequation,whichbelongs to the classofparametricallyexcitedvibrations.The test errorsof thebest runs areplotted inFigure8.7 as functionof their computationtimes. The errors are the minima of the NMAE values of the 25 runs for the vibrationsmentioned in Table 8.1. There are two outliers: the negative damping evolved with apolynomialtermtookmuchlongertocomputeandthebestrunfortheMathieuequationissignificantly lessaccurate. It isremarkablethatthebestnon‐lineardampingrun isslightlybetterthantheothernon‐linearcases.
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Figure8.7.Thebestachievedtesterrors(NMAE)outof25runsasfunctionofthecomputingtimeofthebestruns for thecases listed inTable8.1.Thenumbers inside the figuredenote thenumberofevolvedcoefficients.8.4.3 SensitivityanalysisAsensitivityanalysiswasdone tostudy theeffectofdifferentpopulationsizes,numberofgenerationsandtolerancesettingsoftheODEsolver.TheresultsaresummarizedinFigure8.8.ThetesterrorsareNMAEvaluesover25runs,asdefinedinequation8.1.Thesensitivityanalysis isbasedonthethree‐dimensionaloptimizationproblemoffindingthecoefficientsofanunforcedvibrationwithnon‐lineardampingterm‐0.4501+1.0283y+1.903y2.
Figure 8.8. Sensitivity analysis results. Left: sensitivity on population size and number of generationsshowingnormalizedmeanabsoluteerrors(NMAE)of25runs.Right:sensitivityoftesterror(NMAE)ontolerancesofODEsolvers forcomputing fitnessvalues(“evaluationtolerance”)andtesterrors(“testingtolerance”).Ontheaxes,“1e‐2”means10‐2,etc.andthegreyscalereferstobase‐10logarithmsofNMAEvalues.
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PopulationsizeandnumberofgenerationsTheresultsshowthatapopulationsizeof50yieldsfarbetterresultsthanapopulationsizeof 25 (Figure 8.8, left), and 100 generations score far better than50 generations. Furtherincreasesinpopulationsizeandgenerationsgiveconsiderablysmallerimprovements.SolvertoleranceTherightplotofFigure8.8showstheeffectofdifferentcombinationsofterminationsettingsfor the numerical integration algorithm used for computing errors of the candidate ODEmodels “evaluation tolerance” and of the winning model “testing tolerance” .Unsurprisingly, stricter lower tolerances lead to more accurate results “‐4” in thecolourbarreferstoaNMAEvalueof10‐4,etc. .Theworstresultsoccurforastrictevaluationtolerance in combination with a coarse testing tolerance. Furthermore, it is seen thatrelativelylowtesterrorsarefoundiftherelativetoleranceoftheODEsolverforevaluationand testing are the same. Of course we need to realise that the lower the evaluationtolerance, the longer on average the runs are likely to be and that the testing toleranceshould always be relatively strict in order to make a fair judgment. Based on theseobservations, a relative evaluation tolerance of 10‐3 and a testing tolerance of 10‐5 werechosenforthesimulationsintheprevioussections.SolveralgorithmApartfromthechoiceofresidualerrorasamodelparameter,asuitablechoiceofintegrationalgorithm is also paramount accuracy and runtime. MATLAB suggests the use of specificbuild‐inODEsolversforstiffproblems(MathWorks,2010).Quateronietal. (2010)discussthe application of different ODE solvers in MATLAB. From this, stiff solver ‘ode15s’ wasselected to compete with ‘ode45’, the standard solver that was used in all work in thischapter and the next. The Van der Pol oscillator was used as test problem: increasing μvaluesmaketheproblemstiffer.Theresultsof25runsforeachsolverareplottedinFigure8.9.
Figure8.9.ComparingODEsolversofMATLAB.Left:NMAEtesterrorsover25runsplottedasafunctionoftheμparameterofaVanderPoloscillatortestproblemwithtwoconstantsinthenon‐lineardampingterm.AllothersettingswereidenticaltothemainrunsinSection8.4.2.Right:thestandarddeviationoftheseerrors.
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Theresultsindicatethatforμ≤1.0,bothsolvershavemuchlowererrorsandsmallerspreadthan forμ >1.0.Moreover, the stiff solver ‘ode15s’ has lowermaximum test errors andasmallerstandarddeviationoferrorsforμ>1.0.Itwasindeedexpectedthatthestiffsolverperforms relativelywell for increasingμ. This probably becomes evenmore apparent forstiffertestproblems.NexttotestingtheODEsolvers,thiswasofcourseatthesametimeatestfortheDEalgorithmtosolveincreasinglyhardoptimisationproblems.8.4.4 ImprovingthesearchbyincludingresultsfromFFTAnideaputforwardbyHowardandOakley(1994)istouseinformationfromthefrequencydomainofthetargetdataintheevolutionarysearch.Thissectiongivesabriefexplorationofthis idea within the context of the preceding numerical experiments. The spectralinformationcanforinstancebeappliedtointroduceabiasattheinitialisationorduringthesearch. Let us have a look how the period and stability of the total system relate to thecoefficients.Thefollowingnon‐linearODEwithapolynomialdampingfactorisconsidered:
1 0, (8.6)
whichincludestheVanderPoloscillatorasaspecialcase.Afewdefinitions:
0 ⁄ undampednaturalradialfrequency,asbefore1 naturalradialfrequencyofthesystemasawhole1 1 0⁄ standardizednaturalradialfrequencyofthesystemasawhole
In equation 8.2, k =m = 1, such that for this system 0 1 and 1 1. If a one‐to‐onerelationbetween 1andthecoefficientsμ,νandρwouldbefound,thenmeasuring 1wouldimmediatelygivegoodestimatesoftheirvalues.However,theeffectofnon‐lineardampingon frequencyof free vibrations ismuch less obvious than for non‐linearity in the springs.Twoillustrativecasesaretreated.Case1:ν=0,ρ=1SchmildandGuicking(1980)giveagoodapproximationof 1forthestandardunforcedVanderPolsystem.Figure8.10showsthreeanalyticallyderivedrelationsmentionedinSchmildandGuicking(1980)as lines.NumericalexperimentsweredonebysimplysolvingtheVanderPolequationnumericallyandperformingaFFToftheresult,tofind 1(plottedas+’sinFigure8.10).
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Figure8.10.AnalyticalestimatesofthenaturalfrequencyoftheVanderPolsystem.Thevaluesfromthenumerical experiment were achieved by solving the ODE over a reasonably long time interval andsubsequently performing a fast Fourier transform with triangular windowed smoothing to find thedominantfrequency.Indeed, theanalyticalmodelandtheSchildandGuickingmodelcoincide.This implies thatfor the standard Van der Pol ODE, performing a FFT gives μ directly and furthercomputationsarenotnecessary.Case2:ρ=1Now,amoredifficult case: 1 2 0.Aseriesofexperimentswasdoneforvariousμandν.Figure8.11(left)showsthephaseplaneforthreedifferentνforconstantμ.Figure8.11(right)givestheequivalentofFigure8.10.
Figure 8.11. Non‐linear damping simulations. Left: μ = 3, varying linear term coefficient ν. Right:normalizednaturalfrequencyoftheoscillatorasfunctionofμandν.
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ItisseenfromFigure8.11(right)thatdifferent(μ,ν)‐pairsgiveidenticalsystemfrequency.Theplotshowsthatthesignofνisirrelevantfor 1andhencecannotbefoundfromFourieranalysis,which isalsotrueforthedampingcoefficientc inthestandard linearODE.Thesefindings suggest the following procedure for applying spectral information into systemidentificationcomputationsforthisODE:
‐ FFT on the mildly filtered time signal gives a number of spectral peaks. Take thehighestpeak atnonzerofrequency asthedominantsystemfrequencyandplotthisasahorizontallineintheμversus 1 diagram dashedlineinFigure8.12,left .
‐ It is then deduced from this diagram that the observed frequency could have beengeneratedbyacombinationof(μ,ν)‐valuesaccordingtoallintersectionswiththelineofconstant 1,thislineisshowninFigure8.12 right .
‐ The DE search algorithm should then be adjusted so it biases values in theneighbourhoodofthisline.
Figure8.12.Estimatingcoefficients fromspectral information.Left:plottingmeasured frequency.Right:interpolatedlineofpossible(μ,ν)‐values.Additionally,Itisusefultohaveinformationaboutthestabilityofthesystemasafunctionofthe three parameters, so that for instance candidate models in unstable regions can beskipped.Figure8.13showstwoMonteCarlosimulationsofsolutionstabilitiesforarangeofμandνvalues,keepingρ=1(left)andρ=0(right).Havingthisinformationbeforestartingcouldimprovetheevolutionarysearch.
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Figure8.13.MonteCarlosimulationsofstabilityin(μ,ν)‐planeforρ=1(left)andρ=0(right).TheODEwas integratedovertenperiodswitharelativesolveraccuracyof1e‐3(‘ode45’solver).Lightgreydots:fulltimecomputed(stable),blackdots:solverfailedorsolutionwasoutofsetboundsbeforeendtimewasreached (unstable).Varying initialconditionsdidnot influence theregionboundaries.6000 trialsweremadeforbothplots.Numericalobservationsofstabilitysuchasthesearecertainlynotimpossibletoexplain,butgenerallytheanalyticalanalysesbecomeinconvenient.Surely,naiveintuitionstillhassomevalue in relatively easy cases: for ρ = 0 (Figure 8.13, right), the damping term reads– andonecouldhaveguessedthat,roughly,μ>0givesunstableandμ<0givesstable solutions. If the damping term would have had the form | |, | | , with F apolynomial, thenapositive constant termof thispolynomialwouldalways implya stable,andanegativeconstantanunstableequilibriumposition.SchmidtandTondl(1986)namethis as a simpleexampleof a self‐excited systemwitha singularpoint in thephase‐plane,which represents a single equilibriumposition. Alternatively, the stability analysis of self‐excited oscillations focuses around (stable or unstable) limit cycles (e.g. Figure 8.11, left).However,mostnon‐linearcasesrequirea largetoolboxofanalyticalmethodsforrevealingstability properties, especially when more than one d.o.f. exists or when external orparametric excitation is added. The textbooks by Schmidt and Tondl (1986) and Verhulst(1996)providebridgestoadvancedliteratureonnon‐linearvibrationanalysis.8.4.5 DiscussionIthasbeenshownthatcoefficientssetsofODEscanbefoundfromtimeseriessuchthattheerrorsbetweentargetandmodelaresmall.But todetermineactualvaluesof independentcoefficients,knowledgeofthephysicaldomainisrequired.Thiscanbedoneinvariousways,e.g. solving 0asaconstrainedoptimizationproblembyprescribingupperand lowerbounds form andk; orby solving 0with 2 and and applying available knowledge on themass afterwards. All knowledge and estimationtechniquesfromSection2.4arewelcomedforformulatingconstraints.Thegoalistoletthesystemidentificationfocusonfindingthetoughestphysicalquantities.Structuralmassandstiffness are obviouslymore easily estimated frommechanical pre‐studies than hydraulicdampingisfromahydrodynamicpre‐study.SeealsoSection8.6.
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TheODEsolvertestshelpedtorevealthedilemmaofusingdifferentintegrationschemesforspecificproblems,versusmaintainingrobustness–atthecostof longercomputationtimesorlossofaccuracy.Toocoarsesolvershavetheeffectthatpromisingcandidatesdonotcomeout on top. The next chapter attempts tomake a first step at automated customisation ofsolveralgorithms.
8.5 Applicationofevolutionarysystemidentificationtoexperimentaldata
The experimental data set analysed in Chapter 5 consists of response forces for quasi‐stationary hydrodynamic situations of constant discharge, water levels and mean gateopening,andwherea certaindynamicequilibriumbetweenhydrodynamicsandstructuralresponsehasbeenachieved.Theabsenceofexcitationsignalsofpressuresactingonthegatemakes ithardor impossible formanytechniques to identify thesystem. Ideally,wewouldliketofindthefullmotionequation,thedisplacementODE,fromthemeasuredforcesignal.ConforthandLipson(2013)discusspossibilitiesandexamplesofinferringODEsforsystemswithhiddenvariables,butdonotgivehelpfultipsforthepresentcase.Looking at the acquired quasi‐stationary signals, these can roughly be divided into threegroups: low‐level noise, regular high‐level amplitude oscillations and transitions betweenthesetwo.Theirregularnoisysignalssometimesdisplayvibrationsatthenaturalfrequency,butthesetypicallydieoutagainafterafewseconds.Thismeansthatthepresent(external)excitationisdamped.Intheinterestingtransitionalzone,occasionalhigheramplitudesfeedenergy to the oscillation. Here it seems that the damping consists of both positive andnegative contributions,whichmay varywith time and/orwith displacement (amplitude).Then,atlargeamplitudes,thesineshavebecomeveryregular;preciselysufficientstructuraldampinghasbeenmobilisedtobalancetheself‐excitationforcessothattheamplitudesareasgoodasstable.For system identificationonly the transitional signalsare interesting.Thechallenge lies inthe fact that the damping force is always small compared to the other forces; and thuspotential damping non‐linearities appear only as very slight deviations from the standardsine.Moreover,non‐linearityintheinertiatermduetoaddedmasseffects(whichareusuallysmall) can cause a distortion. A number of preliminary identification tests on rawtransitional signals with DE based on plausible equation structures (as investigated inSection8.4.2)proveddemanding.The issuewithstudying the irregularities is thata lotofanalysis (many signals,manypartsofone signal) isneededbeforemeaningfulhypothesescomeabout.Instead, an unsteady signal of the same experiment series is used to explain a possibleapplicationprocedure.Thisconcernsmerelyaproofofconcept.Withoutgivingunnecessarydetails for this illustrative example, at a constant installed stiffness thewater levelsweregradually(butunevenly)loweredgivingachangingheaddifferenceandresponsefrequency.As Vr also slowly changed, a vibration region was encountered and vibrations started toappear.TheanalysedpartofthesignalF(t),measuredat200Hz,captures10secondsofagrowthphaseofregularvibrations.TheexperimentaldataisdepictedasdotsinFigure8.14.
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Figure8.14.Partofasupport forcesignal(sameset‐upas inChapter5)underunsteadyhydrodynamicconditions. The points are measurement data, the line is the identified model. The support force iscomputedfromtheequationofmotion,thecoefficientsofwhichwereevolvedwithdifferentialevolution.Themain issue isdealingwiththefact thatthesupport forceF t wasmeasuredwhileweare interested in the motion equation , , . Some trial and error resulted in thefollowingapproach:‐Tofindtheinitialconditions,anundampedforcesignalisderivedwithanamplitudethatisthe average amplitude of the first few periods of the target signal. Assuming c 0 andlinearity, the support force is F kAsinωt for an assumedmotion y Asinωt. This givesrightawayy0 F0/k,withthesubindex0indicatingvaluesatt 0,becausethestiffnesskwasmeasured.Theinitialvelocity andangleareevolvedinaDEpre‐runwithstructureF t C1sin C2t C3 .ThenC3istherequiredinitialangleand ⁄ .‐Asa first check, the systemof equations consistingof 0, F ky, y0, canbesolved toretrieveanundamped forcesignalwithequivalentamplitude, thatshouldmatchthefirstpartofthesignalwheredampinghaslittleinfluence.‐ Then, in themain run the evolved initial conditions y0, are used in the evolutionofcoefficientsofODEs , , ,withstructureshypothesizedfromtheory.Anerror‐basedfitnessfunctionisusedbasedonthemeanabsoluteerror:
fitness1
, , , (8.7)
wherethef1‐termistheevolveddampingforceandf2istheevolvedstiffnessforce,Fexpisthemeasuredsupportforceandnthenumberofdatapoints.‐Forthepre‐runandthemainrun,allmeasuredpointswereusedintrainingandhencenotesterrorsarecomputed.Acomparisonoferrors afternormalisation betweenpre‐runandmainrunindicatesoftheconjecturedmodelstructureisanimprovementwithrespecttoazero‐dampingmodel.Thishelpstoconfirmorrefuteanalyticaltheories.Figure8.14showsthebestfittingmodel shownasaline withanon‐lineardampingterm.The plotted line is the sum of the cimputed damping and stiffness terms. Two notable,
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manual additions were necessary to adjust the model structure: the moment that thedamping first becomes negative could not be found, this was therefore added as a step‐functionH t ,andatime‐dependenttermwasaddedtothestiffnesstoaccountforaslowlydroppingfrequencyduetothefallingwaterlevels notvisiblewiththenakedeyefromtheplot .TheevolvedODEisthus:
19.14 1.4 12.18 4.473 ∙ 10 81360 1 2.986 ∙ 10 1.4 0,
with 0.3368 mm and 5.8 mm/s. (8.8)
In this expression, only the installed stiffness 81360 N/m has been used as domainknowledge. The only applied pre‐processing was subtraction of the moving average. Thecoefficientsarephysical;19.14isthetotalvibratingmassinkg,thenegativehydrodynamicdampingtermis‐12.18Ns/mandtheextrastructuraldampingenteringathighamplitudeis4.473 107Ns/m3.Howsensiblethesevaluesareisimpossibletosaywithoutfurtheranalysisor comparison with other data, because all knowledge of the experimental conditions isalready used. The time‐varying stiffness is perhaps unlikely physically; the frequencyprobablychangesviatheinertialterm.Thesetwotermsarenotoriouslyhardtodistinguishsincetheyareinphase.Itwaschosentousethestiffnesshere,becauseitrelatesdirectlytothemeasuredforcesignal seeremarkbelow .Aniceconsequenceofthemethodisthatahysteresisplotcanbemade,seeFigure8.15.Thedotsaremodelleddatapointsoftheoscillatingforceonthemass.Normallytheellipticareagivesthehysteresisloss workdonebydamper ,buthereitrepresentsenergyfedintothesystem.
Figure8.15.A ‘deflectionversusforce’plot(zoomedin)showingmodelleddatapointsofthevibrationinthepreviousfigure.Theareaenclosedbytheellipse(nolineswereplottedbetweendotsforclarity)equalstheenergyfedintothesystem.Lastly, tworemarksabouttheapplicationofDE.First,someassumedmodelstructuresaremuchhardertocompute,forexamplewhenasign( )‐termisincludedforCoulombdamping.
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Typically,astandardrunof80generationswithapopulationof35tookroughly20minutes.Secondly,evolvingcoefficientsthatareindirectlyinvolvedinthecomputationofthefitnesswasfoundverytricky.Forinstance,theinitialconditions,oratime‐dependentexternalforceterm in the motion equation are practically impossible to evolve when the fitnesscomputationusesdampingandstiffness forces.Suchtermshave tobe found inadifferentway–herethesolutionwastoassumeunforcedself‐excitationandfindtheinitialconditionsinapre‐run.
8.6 Geneticprogrammingandsymbolicregression
In genetic programming (GP), introduced by Koza (1992), the individuals are executableprograms represented by tree structures (parse trees) and dedicated to solving complexproblems. It canbe applied very effectively to symbolic regression (SR), a formof systemidentification aimed at finding analytical expressions, for example 0.25x2–sin(3.6x), todescribetrendsinanumericdataset.WhatmakesSRsoappealingisthatitsimultaneouslystrives to find coefficients and model structure with a minimum of presupposed domainknowledge.SymbolicregressionhasgrownfromabenchmarkfortestingnewGPtechniquesto a competitive tool in scientific computation. The general caution in regression analysisandsystemidentificationthatachoiceforaparticularnon‐linearmodelstructureisdecisivefor performance does not hold for GP: the freedom of the tree representation enablesextreme flexibilityandmoreovercontains linearmodelstructuresasspecialcases (e.g. forpolynomials,a1x+a0 isaspecialcaseof∑ ). InnumerousauthorshavecontributedtothedevelopmentofGPinthe1990sand2000s;Polietal.(2008)servesasanintroductiontoliterature.A recurring point of attention in GP is ‘bloat’, defined as offspring growing inmodel sizewithout clear benefit in termsof fitness (Poli et al., 2008), canbe controlledby somehowincludingthelengthofcandidatemodelsintheevolution.TheapplicationofGPtoSRbringsadditional challenges. To name two major subjects, hitherto without universally appliedsolutions:determiningnumericalconstants(i.e.coefficients)anddealingwithnoisydata.Itwas originally proposed to let numerical constants evolve in the same way as theindependent variables. This proved to be an inadequate approach in practice. Among thenumerousimprovementsistheuseofdifferentialevolutiontofindtheconstants(Cernyetal.,2008).Noiseisanindigenoustopicinmachinelearning;itiswellknownthatdeterminingnumerical constants from noisier training data increases the ambiguity of model findingsincetherewillbemorecombinationsofstructureandconstantspossiblewithcomparableaccuracies (Rogers andGirolami, 2012).Generalizationpropertiesof the finalmodel areakeyissue,thisisknownasthebias/variancetradeoff(Bishop,2006).InGP,thepresenceofnoise tends to promote complexity of the evolved programs and bloat (Zhang andMühlenbein,1995;KeijzerandBabovic,2000).Theproblemofbalancingthesearchforafitmodelwithanacceptablecomplexity(modellength,numberofterms)isreferredtoinGPas“accuracyversusparsimony”.The fact that GP‐based SR is more than a data regularization technique and exceeds thepossibilities of preceding system identification approaches (Conforth andLipson, 2013) isreflected in the term ‘knowledge discovery’. The training data is not just used to infer anumericalmodel,but toderiveusabledomainprinciples inconcisemathematical form;allimaginableclosed‐formequationsandordinaryandpartialdifferentialequations.Thisputs
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thedichotomybetweenwhite‐box andblack‐boxmodelling on its head.Anumber of PhDstudieshaveworkedonSRbyGP, fromacomputerscienceperspective (Keijzer2001andKromberger 2011) and from amore applied perspective (Vladislavleva 2008). An elegantdemonstration of the power of GP for identifying non‐linear dynamical systemsautomaticallyanddirectlyfromexperimentaldatawasgivenbySchmidtandLipson(2009)in a seminal double pendulum experiment. A consequence of this work and the relatedcommercialsoftwarepackageEureqa(Eureqa,2011)hasbeenanexplosionofapplications.The system identification efforts in this chapterwere limited to searching for coefficientsafterfixingthemodelstructure,aso‐calledparametricapproachtoregression,accordingtoVladislavleva (2008). The goal of extending the evolutionary algorithm with GP –not forfindingtheODEstructure,butfordeterminingthenon‐linearterms–wouldnaturallybetodiscovernewmodelstructuresintangibletoanalysisfromprinciples(becausetheyarefar‐fetched,ortheinvolvedanalysiswouldbetootime‐consuming).Aspartofthisstudy,aGPcodewaswritten using gene expression programming (GEP), a GPmethod introduced byFerreira (2001), and tested on a number of benchmark regression problems. It was notapplied to the experimental data, because thiswould enable an analysis of amaximumofonly two seconds with long computation times. Also, some trials were donewith Eureqa(Eureqa,2011).Therewasnooptionheretoinferthemotionequationfromtheforcesignal.So this resulted in rather hard‐to‐interpret expressions (depending on selected buildingblocks) of F(t) and , , , requiring about half an hour to one hour for threesecondsofdata.
8.7 Conclusionsandoutlook
In thischapter itwasexaminedhowthedifferentialevolutionalgorithmcanbeapplied toidentifyseveralvibration typesbyperformingregressionon thecoefficientsof themotionODE. Irrespective of the percentage of data used in training, an ODE structure producedmoreaccurateresultsthanananalyticalsolutionstructureofaforcedvibration,butrequiredmorecomputationaltime.Anumberofsyntheticself‐excitedoscillationswasidentifiedwithreasonable accuracy. The presence of superfluous non‐linear terms proved to have aninfluence on the achieved computation times, but not directly on test errors. SensitivityanalysesexposedtheimpactsoftolerancesettingsoftheODEsolverandthechoiceofsolveralgorithm itself.Thesearchcouldbenefit fromspectral informationof theoutputsignal toreduce the search space, but (the success of) this process depends on the investigatedequation structure of the non‐linear terms. The tests showed running times that are atpresenttoohighforquickassessmentapplicationsinearlywarningsystems.Speed‐upcanbeachievedbyfastercodingandparallelization.Twoideasforfuturepossibilitiesaregiven.OntheonehandGP‐basedSRisarevolutioninhowmeaningfulmodelsarederivedfromdata,ontheotherhandthisautomateddiscoveryinnowayexemptsscientistsfrommakingthoroughinterpretationsoftheresults,asalreadynotedbySchmidtandLipson(2009).
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Figure8.16.Theaccuracy/complexityfront,aplotfromSchmidtandLipson(2009).Theresultofanevolutionarysearchforanalyticalmodelsisanaccuracy/complexityParetofront.SeeFigure8.16;SchmidtandLipson(2009)refertothisas“predictiveabilityversusparsimony”.Theysettlethistradeoffbysuggestingthemodelattheinflectionpoint,i.e.rightafter the accuracy has made a jump and before the front’s tail has a steep increase incomplexityachieveslittleextraaccuracy.Itwouldbeinterestingtoseeifthischoiceisinfactthe best, for a variety of applications, in terms of generalisation and insight. Whatcomputational tools can assist inmaking this choice and does the evolution benefit fromthis?Asecondfuturequestion:whatistheminimumamountofdata(fromphysicalornumericalexperiments)necessaryforautomatedderivationofthefullsystemofequationsofthefluid‐solidinteraction,i.e.theNavier‐Stokesandstructuralequations?Afirstanswerwouldbetofeed the systemwith time series of flow velocity and pressure, at the interface and otherstrategic locations, that representdiversehydrodynamicconditions. Inparticular, thedatashouldsufficientlycoverdifferentranges(flowregimes)of theReynoldsnumber.Trainingthe model to recognise and describe different Reynolds number flows will be verychallengingandrewardingatthesametime.Apartfromthetheoreticalimpact,thiswillfindapplicationsinturbulencemodellinginCFDande.g.inmodelsforanomalydetectioninearlywarningsystems,notunlikethecontrolsystemofChapter7.
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9 Inferringnumericalalgorithms8
9.1 Introduction
AnobviousdrawbackofthederivationofODEsinSection8.4isthatcandidatemodelsneedtobesolvednumericallyinordertobeevaluated.Itwaschosentocomputeerrorswiththeoriginal target data series y(t) and the predicted time series . For a first‐order ODE
, onecouldevolvetheright‐handsideandcomputeerrorsbycomparisonwiththederivativeofthetargetseries.Reducingan‐th‐orderODEstoasystemoffirst‐orderODEs,then‐thderivativeofthetargetdata isneeded.Differentiationof theoriginaltimeseries–especiallywhendonemorethanonce–isnotatrivialtaskduetothenoisepresentinreal‐lifedata (seeKronberger2011). In theprevious chapter’s study thevelocitywas requiredonlyatthetimeofthefirsttrainingpointasinitialvalue.Other approaches assume structures that are easily computable, such as thediscretemapused by Howard and Oakley (1994), but these often provide less insight in the systembecausetheresultingexpressionisunfamiliarandthereforehardtointerpret.Inanattemptto have the best of bothworlds, one interesting alternative9 is to perform SR on discreteequations and afterwards derive the continuous‐form ODE from the discrete expression.Continuing with the example of finding a numerical solution to a second‐order ODE, thiscorrespondstothesearchforadiscretisedversionofanODEthatlivesinthesolverdomainasinFigure9.1.
Figure9.1.InferringODEsatthediscretelevel.ThepartenclosedbythedashedlinerepresentsunknowndiscreteformulaewhicharesolvedbyevolutionarycomputingandafterwardsconvertedtotheODEform.Thecentralideathatmakesthisworkisthatthecontinuousform,i.e.theright‐handsideof
, , is always derivable from the recurrence relation form by differentiation withrespect to the timesteph and taking the limith→0. Basic integrationmethods for solving
8 Parts of this chapter are based on the manuscript “Evolutionary design of numerical methods:generating finite difference and integration schemes by differential evolution” by C.D. Erdbrink,V.V.KrzhizhanovskayaandP.M.A.Sloot.9 This idea was put forward in a presentation by Maarten Keijzer during the symbolic regressionworkshopofGECCO2013,themainconferenceonevolutionarycomputing.Atthattime(July2013)ithad not appeared in publications and so far no publications were found that elaborate on thisapproach.
ODEy (t)
initial conditionsODE solver algorithm
(y0, y0)known
unknown
known
f( t, y, y, y ) = 0
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initial value problems (Euler, Heun, etc.) all have the form , , , for acertain function F. Suppose an expression like this is derived through evolutionarycomputing, then f(t,y) is found fromlim → ⁄ .Thiscouldprovidenewcomputationalpossibilities, provided that both the variation operations and fitness evaluation of thediscreteformcanbeprogrammedefficiently.
9.2 Inferringsolveralgorithms
9.2.1 IntroductionTheODE solution process can have yet another unknown: the solver algorithm itself. SeeFigure9.2.Thenewideaintroducedinthischapteristouseevolutionarycomputingtofindformulaeor ‘schemes’forsolvingnumericalproblems.Theconceptofcontrivingequationswithouthumaninterventionisthusappliedtonumericalmathematicsitself.ThisisareversemodellingproblemwheretheODE,theinitialconditionsandthesolutionareassumedtobeknown.
Figure9.2.Findinga solveralgorithm foranODEwithaknown solution.This isamodellingproblem,wherethestructureand/orthecoefficientsofthenumericalsolverschememaybeunknown.Inthissection,coefficientsofclassicalnumericalschemesfordifferentiationandintegrationare generated by an evolutionary algorithm (EA). EAs generally become interestingwhenapproximate solutions are acceptable and small improvements over existing solutions areextremely valuable. In addition, it is required that testing of the quality of candidatesolutions should be unambiguous. The problem at hand, finding discrete computationalschemes for estimatingderivatives and integrals,meets these conditions: inexact schemesareacceptableastheschemesthemselvesrepresentapproximations,andtheperformanceofa scheme is easily checked by applying it to a function with known properties. Thewidespread use and therefore importance of finite difference methods and Runge‐Kuttamethods is self‐evident. Basic finite difference schemes have straightforward analyticalderivations,butcomplexcomputationalscienceproblemsoftenrequirethedesignofmoresophisticatedmethodslackingstandardapproaches.Runge‐KuttamethodscompriseafamilyofintegrationschemesforsolvingODEinitialvalueproblems, themost famousbeing the fourth‐order (RK4)Runge‐Kuttamethod. Fororderspast two the systems of order condition equations are underdetermined, this hampersanalyticalderivation.TsitourasandFamelis(2012)notethatoptimizationmethodsbasedoncomputing Jacobians are less suitable for finding Runge‐Kutta schemes than EAs, sincedetermining the derivatives of the variables as they appear in the order conditions iscumbersome.
unknownknown
y (t)initial conditions
ODE solver algorithm
(y0, y0)
known
known
f( t, y, y, y ) = 0ODE
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Forthemodellingtaskofgeneratingbothascheme’sstructureaswellasitscoefficients,theparse tree method of GP provides an appropriate approach. However, if the equationstructure is given and the goal is to find only the coefficients, it becomes an optimizationproblemandotherEAscanbeused.Here,againdifferentialevolution(DE)willbeused.ThepowerofDE is reflectedbyoutperforminganumberof stochasticoptimizationalgorithmssuch as Adaptive Simulated Annealing (Storn and Price, 1997) and Particle SwarmOptimization(VesterstrømandThomsen,2004); itbeatsGeneticAlgorithms(GA)onmanybenchmarksaswell(TušarandFilipič,2007;Hegertyetal.,2009).Previously,theapplicationofEAstoevolvecomputationalalgorithmswasmostlyaimedatsolving domain‐specific problems. For example, Spector et al. (1998) used GP to findquantumcomputingalgorithmswithsuperiorperformance.AnumberofpaststudiesutilizeEAs inmore general contexts to solve systems of equations or somehow tune traditionalnumericalmethodstoimproveperformance.Forexample,Heetal.(2000)proposeahybridalgorithmthatcombinestheclassicalSuccessiveOver‐Relaxationmethod(SOR)forsolvinglinearsystemsofequationswithanevolutionarytechniquetoevolvetherelaxationfactor.A study by BaniHani (2007) uses GA to determine points and weights for an integrationprocedurewithin ameshfreemethod for solvingboundaryvalueproblems.To thebest ofourknowledge,MartinoandNicosia(2012),TsitourasandFamelis(2012)andNanayakkaraetal.(1999)aretheonlyeffortssofartoapplyanevolutionaryalgorithmtoderiveRunge‐Kutta‐relatedmethods.ThefirststudyadvocatestheuseofEAsinnumericalanalysisfromtheviewpointofalgebraicgeometry,butdoesnotpresentconcretevaluesofcoefficients,nordoesitdiscussaccuracy.ThesecondstudyconsidersneuralnetworksandDEtofindRunge‐Kutta‐Nyströmpairs forsolvingasecondorderODEproblemoccurring inastronomy.Thethird study uses an EAwith the purpose of finding shape parameters of Runge‐Kutta‐Gillneuralnetworks,throughradialbasisfunctionnetworks,fortheidentificationofrobotarmdynamics.In the field of finite difference methods, an effort by Haeri and Kim (2013) uses GA tooptimize boundary characteristics of compact finite difference equations. These recentstudies illustrate the relevance of applying EAs to find involved numerical recipes andprovide the motivation for our research, where we apply DE for deriving general, i.e.domain‐independentnumericalschemes.The aimof this chapter’s study is to apply differential evolution to find coefficient sets ofclassicalnumericalschemes forapproximatingderivativesand integrals.Thiswillnotonlyproduce new coefficients for schemes where fully analytical derivation of coefficients isunattainable, but the reverse‐engineeringprocesswill also yield insight intohow toapplythe evolutionary heuristic andwhat the resulting accuracy is. This paves theway for theapplicationofEAstomorecomplexnumericalstencilsofgeneraluseinappliedmathematicsandengineering.The next subsection treats related work. Then a background on the studied numericalschemesisgivenanditisexplainedhowthecoefficientswillberepresented.Thisisfollowedbyadescriptionofthecomputationalmethodusedinthisstudy.Thenthemodellingresultsare presented; this subsection consists of a sensitivity analysis and presentation and
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discussionof trainingandvalidationresults. InSection9.3, conclusionsaredrawnand9.4givesareflectiononthischapterandthepreviousonewithanoutlookonfuturework.9.2.2 Background1.FinitedifferencederivativeestimatesCommonfinitedifferenceformulaeareof thetypecentral, forwardorbackward.Table9.1gives two examples, viz. the second‐order central formula and the second‐order forwardformulaforestimatingthefirst‐orderderivative.Table9.1.Representationoffinitedifferenceschemes.
finitedifferenceformula vectorrepresentation
2 1/2 0 1/2
3 4 22
3/2 2 1/2
Herehisthestepsize.Byputtingallcoefficientsinthenumerator(thusleavingonlyhinthedenominator)eachapproximationschemeisdefinedbyitscoefficients ∈ ℚforallterms
∙ with ∈ .Thenumberof termsgrowswith increasingorderofaccuracyand the coefficients can, among other ways, be determined via the Taylor series. Noteadditionally that the same coefficients used in a forward scheme can also be used in anequivalentbackwardschemewithsymmetrictermsf(x±h).Forevenorderderivatives,thecoefficientsareequalforcorrespondingterms,butforoddorderderivativesthecoefficientsaremultipliedby‐1.Acentralschemeoforderpcontainspnonzeroterms,wherepiseven.Thepcoefficientsofacentral schemewill be searched for by evolving vectors of sizesp andp+1. The formula’sskeletonassumedforrunswithvectorsofsizepconsistsofterms withq=1,2,..,p/2 and vectors of size p+1 contain the extra term . A forward scheme of order pcontains p+1 nonzero terms andwill be represented only by vectors of size p+1. For theforwardschemetheskeletonconsistsoftheterms withq=0,1,..,p.2.Runge‐KuttaschemesRunge‐KuttaschemesforsolvinginitialvalueproblemsaresummarizedinButchertableausor arrays (Butcher 2008). Table 9.2 shows this notation and also gives the correspondingvectorrepresentationthatisusedinourstudy.
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Table9.2.RepresentationofexplicitRunge‐Kuttaschemes.
Butchertableau vectorrepresentationofexplicitscheme
Whereciarecalledthenodes,wiaretheweightsandsisthestage.TheButchertableauholdsthecoefficientsfortheformula
with ; . (9.1)
ForexplicitschemesthematrixAcontainingentriesaijislowertriangular,i.e.aij=0fori≤ j.Furthermore, the consistency condition requires that the sumof the row elements inA isequal to the node at that level: ⋯ , ∀ (Hairer et al. 1993). Assumingconsistency therefore implies that explicit schemes canbe encodedby the vector given inTable9.2.Forinstance,a4‐stageschemeisrepresentedbythevector(a21a31a32a41a42a43w1w2w3w4)T.The stage s of a scheme fixes itsmaximum order of accuracy. Order condition equationsdetermineifanorderisactuallyachieved(Butcher2008).Forexample,a2‐stagescheme(s=2)hassecondorderaccuracyifw1+w2=1andw2a21=½.TheorderconditionsuptoorderfiveareincludedinAppendixI.Notethatthe2‐stageschemehaseasyclosure:chooseavaluefora21andtheothercoefficientsare fixed.Bycontrast,higherorderschemesarefarmoredemanding;6‐stageschemesneedtomeet17orderconditionstoattainorder5.Thisstudyconsiders3‐,4‐and6‐stageschemes,reachingamaximumorderof5.3.Adams‐BashforthschemesTheAdams‐Bashforthmethod is an explicit linearmulti‐step integrationmethod (Butcher2008).Dependingonthetargetorderofaccuracy,acertainnumberofprecedingpredictionvaluesareusedinthenewestimate.Thegeneralformulaforak‐termmethodreads
, . (9.2)
c1 a11 a12 … … a1s
c2 a21 a22 … … a2s
…
…
…
…
…
… … . . …
cs as1 as2 … ass‐1 ass
w1 w2 … … ws
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Themaximumachievableaccuracyorderofak‐termmethodisk.Theorderconditionscanbe used to find analytical values of the coefficients. The coefficients representation isstraightforwardfortheAdams‐Bashforthschemes:eachvectorcontainsall for1≤i≤k,forcomputingaschemeofintendedorderk.9.2.3 MethodforgeneratingcomputationalschemesbyanevolutionaryalgorithmTheoriginalDEalgorithm,calledDE/rand/1/bin(StornandPrice,1997),assumesconstantstrategyor‘control’parametersCRforcrossoverandscalingfactorFformutation.Withtheadvent of dynamic parameters adjustment (Eiben et al. 1999), later versions of DE havecontrolparametersthatarevariedaspartoftheevolution(Brestetal.2006).OurstudyisbasedonDE/rand/1/binandadopts theself‐adaptivemethodbyChoietal. (2013)whereeach individual carries its own pair of control parameters. Updates take place everygeneration after the selection by taking the average values of the control parameters ofindividuals that evolved successfully in the past generation and then adding variation totheseaveragesbydrawingfromaCauchydistribution,seethepseudo‐codeinChapter8.The initial population consists of floating‐point vectors with uniformly random entriesbetween ‐1 and 1, but no value constraints are imposed during the evolution. In everygenerationfiverandomlychosenindividuals(otherthanthehighestscoringindividual)arereplaced by completely new individuals, ameasure intended to ensure a healthy balancebetween population diversity and exploitation. Admittedly, later tests showed that thismeasurehadmarginaleffectondiversitybecausethenewindividualsquicklyadapttotheenvironment.Thealgorithmhasfourkeymodelparameters:CR0,F0,populationsizeNPandthenumberofcomputedgenerations,wherethezeroindicesdenotetheinitialvaluesofthecontrol parameters applied to all individuals. Because of the random nature of the self‐adaptation,these initialvalueshavenegligible influenceontheresultoftheevolution;thisstudyusesCR0=0.25andF0=0.6.Thepopulationsizeandnumberofcomputedgenerationswillbedeterminedinasensitivitystudy.1.FinitedifferencederivativeestimatesTrainingconsistsofcomparing firstorderderivativeestimatescalculatedby thecandidateformula with analytical values sampled from a target derivative function. A bell‐shapedcurveanditsderivativeareusedastargetfunction,seeFigure9.3.
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Figure9.3.FunctionpairusedfortrainingcoefficientsoffinitedifferenceschemesandAdams‐Bashforthschemes.ThetargetfunctionpairofFigure9.3is 1.5 . and ′ 1.5 . ,for‐4<x< 4, with h = 0.01. The fitness functionmeasures the absolute errors between candidatevaluesandanalyticalcoefficientvalues:
log | ∗| (9.3)
Wherev∈ is thevectorcontainingthetheoreticalvalues,v*∈ isacandidatevectorandn = dimR( ). The logarithm is applied to dealwith the large range of values. In thischaptertheevolutionseekstominimizethefitnessfunctions.2.Runge‐KuttaschemesThe candidate coefficient vectors are evaluated by substitution into the order conditionequations.ThefitnessofthecandidateRunge‐Kuttaschemesisthereforedefinedby
log | ∗|#
(9.4)
whereCkistheanalyticalvalueoforderconditionequationk;thesearetheright‐hand‐sidevaluesinAppendixI. ∗ isthecandidateorderconditionsvectorcomputedbyinsertingallnecessaryentriesofthecandidatecoefficientsvectorintotheorderconditionequations.#Cisthenumberoforderconditions.3.Adams‐BashforthschemesFitness evaluation is defined identically to the finite difference schemes, but now thecandidateschemeisevaluatedbyintegratingf ’(fasdefinedinFigure9.3)andtheresultiscompared with the analytical values sampled from f. The necessary starting points are
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generatedbyaRunge‐KuttaschemeofthesameorderoftheAdams‐Bashforthschemethatisaimedfor.9.2.4 ResultsanddiscussionSensitivityanalysisBeforestartingthetrainingprocessofthefinitedifferencerunstheimpactofthenumberofgenerations and the population size are considered, as well as the characteristics of thetarget function.The target functionpairwas chosen in suchaway that thederivativehasboth positive and negative values. Preliminary runs for the sixth order central differencescheme showed that the exact shape (periodicity, number of extremes, symmetry, etc.) oftargetfunctionfhasnonotableinfluenceonaccuracy,aslongasf ’ iscontinuousandhasareasonable variation such that the fitness can discriminate between good and badapproximations.Clearly,thefunctionpairf(x)=4x+7,f’(x)=4wouldbeunsuitable.Varyingtherangeoffunctionf(Figure9.3)between0.1and1000foranequalsamplingstepsizehandequalnumberoftrainingpoints,givingavariationintherangeoff ’between0.4and4850,yieldednodifferencesinachievedaccuracy.Next,thestepsizewasvariedwhilekeepingthetotalnumberoftrainingpointsthesame.Thisalsohadnonoticeableimpactonaccuracy. Figure 9.4 shows the results when the reverse is done: varying the number oftrainingpointswhilekeepinghconstant,forthreedifferentvaluesofh.
Figure9.4.Sensitivitytotrainingdataforthecentralsixthorderapproximationofafirstorderderivative.Left:varyingnumberoftrainingpointsinthetargetfunctionwhilekeepingsamplestepsizehconstant,forthreedifferentstepsizes.Right:sumofabsoluteerrorsasfunctionofthenumberofpointevaluations.Thetwoplotsarebasedonthesame21runswithapopulationsizeof150.Ontheverticalaxesthesumofabsoluteerrorsoverallcoefficientsisplottedonalog‐scale.
Figure9.4(left)showsthatthenumberoftrainingpoints, i.e. thenumberofuniquepointssampledfromthetarget function, isamajor factor foraccuracy.Additionally, itunderlinesthefactthatthestepsizeatwhichthetargetfunctionissampledhasnoinfluenceatall.Theresultsalsoshowthattheimprovementsinaccuracyaregettingsmallerasweincreasethenumberof trainingpoints (N): the sumof errorsgets1000 times lowerwithN increasingfrom1to200;andonly10timeslowerwithNincreasingfrom200to800.
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The number of point evaluations is the product of the number of computed generations,population size and number of calls to training points. The sum of absolute errors as afunctionofnumberofpointevaluations(Figure9.4right)displaysasimilarrelationas forthe dependence on the number of training points, which shows that usingmore trainingpoints does not imply significantly more generations needed to attain convergence.Comparing the two plots of Figure 9.4, we conclude that runs with a similar number oftrainingpointsandadifferentstepsizeachievethesameaccuracy(leftplot)andneedonlyaslightlydifferentnumberofgenerationsforconvergence,resultinginsmallvariationsinthenumberofpointevaluationsperrun(rightplot).The computation time of a run depends linearly on the number of point evaluations andthereforealsoapproximately linearlyonthenumberoftrainingpoints.Theplots inFigure9.4suggestthat improvingtheaccuracyby includingmorethan800trainingpointscomeswithdisproportionallyhighercomputationalcosts.Theplottedtrialrunforthesixthordercentral schemewith 800 training points consisted of about 1.4·108 point evaluations andresultedinasumofabsoluteerrorsof7.8·10‐5.Theimpactofusingdifferentpopulationsizesisinvestigatedbymaking25testrunsforthefinitedifferenceschemes(basedon200 trainingpointsandvaryingbetween50and1000individuals) and for the Runge‐Kutta schemes (using the 6‐stage scheme and varyingbetween50and500individuals).TheresultsareplottedinFigure9.5.
Figure9.5.Sensitivitytopopulationsize.Left: finitedifferencesixth‐ordercentralapproximationof firstorderderivative, changingpopulation sizeNP from50 to1000andkeepingnumberof trainingpointsconstantat200.Right:6‐stageRunge‐Kuttaruns,varyingpopulationsizeNPbetween50and500.Thesumofabsoluteerrorsisplottedverticallyonlog‐scale.The sensitivity to the population size is quite obvious for the finite difference schemes(Figure9.5left):populationsizesof150ormoreoutperformallsmallerpopulationsizes.Forpopulation sizes smaller than or equal to 100, the sum of absolute training errors issignificant–theseschemesnevergiveaccuratederivativeapproximations.Improvementsbyusingpopulationsbeyond150aremarginalcomparedtotheincreaseincomputationalcosts.Averagedoverthreerunsperpopulationsize,thetrainingerrorforapopulationof1000is
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lessthan0.1%smallerthanforapopulationof150,butthecomputationtimeisafactor4.4higher.Usingapopulationof150isthereforeoptimalforthefinitedifferenceruns.ThepopulationsizehasadifferenteffectontheevolutionofRunge‐Kuttaschemes(Figure9.5 right).Whilewinning individuals become exponentially betterwith higher populationsize,thestandarddeviationoftheattainedaccuraciesalsoincreaseswithnumberoffitnessevaluations.Thismeansthatmorerunsarerequiredto improveaccuracyandat thesametimethateachrunbecomesmoreexpensive.Basedonthissensitivity,apopulationsizeof350ischosenfortheRunge‐Kuttaschemes,incombinationwithahighnumberofruns.Thenumberofgenerationsthatmakeuponerunisregulatedbyaterminationcriterion.Theevolutionaryalgorithmstopsifthefitnessofthebestindividualhasnotchangedinapre‐setnumberofconsecutivegenerationsorifamaximumnumberofgenerationsisreached.Bothparametersweredeterminedempirically throughout thesensitivity runsby lookingat theconvergenceof theruns.Figure9.6 illustrates theworkingof the terminationcriterion forthe6‐stageRunge‐Kuttascheme.The‘examplerun’neveractuallyconverges,asitregularlymakes very small improvements but doesnot escape local optima, it stoppedbecause themaximum number of generations was reached. The ‘winning run’ experiences a fewconsiderablejumpsbeforeconverging.
Figure9.6.EvolutionofRunge‐Kutta6‐stagescheme,showingthefitnessesasafunctionofgenerationsforarelativelypoorexamplerunandforthewinningrun(outof100runs).Terminationpointsareindicated.TheplotinFigure9.6alsoclarifiesthatthisparticularproblemdemandsalargenumberofgenerations. The sensitivity runs for Runge‐Kutta were done with a maximum of 75,000generationsandastopafter7,500generationswithoutimprovingthebestfitnessscore.Thiswasextendedtorespectively100,000and10,000generationstoensureconvergenceforthemoresuccessful runs.However, for the lowerorderschemesconvergenceproved tooccurmuchsooner.Table9.3summarizesthemodelparameterschosenafterthesensitivitystudy.
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Table9.3.Chosenmodelsettings.
populationsizeNP
stopafterthis
numberof
generationswithout
improvem
ent
maximum
num
ber
ofgenerations
numberoftraining
pointsN
numberofrunsper
order
finitedifferenceruns 150 250 2,500 800 10Adams‐Bashforthruns 150 250 2,500 6400 10Runge‐Kuttarunsorders≤4 350 500 5,000 ‐ 100Runge‐Kuttarunsorder5 350 10,000 100,000 ‐ 100
The sensitivity analysis reveals a clear difference between the computations based on atargetfunctionandthosebasedonorderconditions.Asdiscussed,fortheRunge‐Kuttarunsthe largevariation inresultsandtheslowconvergencemakes it imperativetomakemanyruns with a high number of generations. Conversely, the finite difference and Adams‐Bashforth runs have quick convergence and moreover attain very similar optima fordifferentruns,sothatasmallnumberofrelativelyshortrunssuffice.Comparedtotherunsof the finite difference schemes, computation of Adams‐Bashforth methods requires lesspointevaluations,sothatmoretrainingpointscanbeincludedinacomparablesimulationtime.TrainingresultsofderivativeschemesThe error used in the fitness computation is the difference between candidate derivativevalues and the analytical values of the target function.However, since the coefficients areknownfromtheoryit ismoresensibletoassessthetrainingresultsbytheabsoluteerrorsbetweencomputedandanalyticalcoefficientvalues.Theseerrorsareplotted inFigure9.7(left).AppendixIIcontainscompletetableswiththeresultingcoefficientsofthebestrunsforallcomputedconfigurations.Figure9.7comparestheresultsofschemesofdifferentorders.Thegeneraltrendisthatthemean absolute error of a finite difference scheme increases with the order. The forwardscheme exhibits a virtually exponential error growth, whereas the central scheme errorsdeteriorateonlypastthesixthorder.Thecentralrunswhereanextraf(x)‐termwasaddedare only slightly worse than the runs where this was not done; the correspondingcoefficientsareclosetozero(seeAppendix II).Figure9.7(left)also indicatesthat thetwospecial schemesattain similar accuracy as the traditional schemes.Also, the results of thecomputed Adams‐Bashforth methods are plotted in the same figure. They are relativelyaccurateandshowanexponentialtrenduptothefourthorder.
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Figure9.7.Left:meanabsolute error of evolved coefficientsas function ofaccuracy order for trainingbasedontargetpointsofthefunctioninFigure9.3.Right:sumofabsoluteerrorsforallhundredrunsforthecomputedRunge‐Kuttaschemesofstages3,4and6.Intheboxplots,thecentrallineisthemedian,theboxedgesindicatefirstandthirdquartilesandthewhiskersextendtothemostextremedatapoints.TrainingresultsofintegrationschemesThedefinitionofthetrainingerrorfortheRunge‐Kuttaschemesisbasedoncomputedandanalytical values of the order conditions. Unlike the analytically tractable finite differenceschemes, the computed coefficients for the Runge‐Kuttamethods do not resemble knownvalues.BoxplotsareshowninFigure9.7(right)foralltherunsperformedforstages3,4and6.Thewhiskersof theboxplotsextendto theextremevalues.Most interestingly, thebestevolvedschemeshaveabsoluteerrors,summedoverallcoefficients,betterthan10‐13forallstages. The variation in results is largest for the 4‐stage scheme, which suggests that inhindsight the termination criterion could have beenmore conservative. Nevertheless, theaccuracyof theevolvedschemesforstages3and4 isboundedbyrounding‐offerrors;theusedsoftware(MATLAB)allowsfor16digitsinthedouble‐precision(64bit)floatingpointformat.Betterresultswouldarguablybepossiblebyusingsymbolicrepresentation,butthisslowsdownthecomputationsconsiderably.ValidationInvalidationtestsweappliedthebestevolvednumericalschemestothedifferentiationandintegration of unseen functions and compared the errors of these numerical solutions(deviationsfromtheanalyticalsolution)totheerrorsoftheclassical(analyticallyderived)numerical schemes. By varying the step size, the slopes of the resulting graphs reveal theorders of accuracy. The Runge‐Kutta schemes are applied to the same non‐autonomousinitialvalueproblemthatwasusedinBoyceandDiPrima(2001)tocomputeerrors:
y 1 4 , 1, with solution14
316
1916
, (9.5)
andallothermethodsareappliedto 2 and ′ 36 .ThevalidationrunsareplottedinFigure9.8forthefinitedifferenceschemesandinFigure9.9fortheintegrationschemes.Inallcases,thenormalizedlocalerrorisconsideredatthesamex‐locationforallschemes.
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Figure 9.8. Validation errors of computed finite difference schemes for approximating first orderderivative. Left: central schemes;Right: forward schemes.The orders of accuracy are indicated in theplots.
Figure9.9.Validationerrorsofcomputedintegrationschemescomparedtoresultsofanalyticallyderivedschemes. Left: Runge‐Kutta schemes; Right: Adams‐Bashforth schemes. The orders of accuracy areindicatedintheplots.Theplotsconfirmthattheintendedordersofaccuracyareindeedreached.Thisfollowsfromthefactthatthereislittledeviationbetweencomputedandanalyticalschemes.Also,itwasverifiedthatthelinesrunparalleltothetheoreticallineshn,forordern.TheplotsofFigure9.8andFigure9.9exhibit twosourcesofdeviation fromtheexpectedstraight line:due torounding‐offerrorsandduetothecomputedschemenotbeingasaccurateastheanalyticalscheme.Thelattererrortypeismanifestedbythebendinglineendsoftheforwardscheme(Figure9.8,right)andtheAdams‐Bashforthschemetowardssmallerstepsizes(Figure9.9,right). The software‐dependent machine rounding‐off error affects both computed andanalytical schemes, as canbe seen forexample for the fourth‐order (4‐stage)Runge‐Kuttaschemearoundh=10‐4(Figure9.9,left).FortheRunge‐Kuttafifth‐ordermethod(6‐stage),the results of the two best evolved schemes are plotted along with the results of a
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theoreticallyderivedscheme(Butcher2008).Remarkably,oneoftheseschemesachievesabettter accuracy than the theoretical scheme: the lowest line in Figure 9.9 (left) has thesmallest errors. This is a coincidental feature related to the choice of the test function.Several additional testswithdifferent functions (not shown toavoid figureovercrowding)exhibitedthesameorderofaccuracywitherrorsfollowingthesameh5trend),buttheerrorswerenotalwayslowerthanthoseoftheanalyticalscheme.
9.3 Conclusions
Thischapterhaslaidafoundationforautomaticderivationofdiscretenumericalschemesbyevolutionary computing. This can become a highly prized approach in situations whereanalyticalsolutionsareabsentandstandardnumericalapproachesfail.It was shown in this chapter how coefficients of widely used numerical methods can becomputed up to practicable accuracies using differential evolution. This was done byrepresenting coefficient sets as floating‐point vectors. Two different training procedureswere applied: based on a target function pair f and f ’ and based on order conditionequations.Finitedifferencemethodsforapproximatingfirstderivativeswerecomputeduptoorder8(centralschemes)andorder6(forwardschemes);alsotwonon‐standardschemesoforder4werederived.Themulti‐stepAdams‐Bashforthintegrationmethodwascomputeduptoorder5.AccuratetrainingofthefinitedifferenceandAdams‐Bashforthschemes,whichemployed samples from the target function pair, was found to depend primarily on thenumberoftrainingpointsandnotonsamplefrequencyortargetfunctionshape.Moreover,there appeared to be a threshold value for the population size required for attainingreasonableaccuracies.Asaresult,thesumofabsolutetrainingerrorsshowedapredictabledecayingexponentialrelationwiththenumberofpointevaluations.ExplicitRunge‐Kuttamethodsweretrainedbypromotingadherencetotheorderconditions,yieldingschemesofstages3,4and6,uptoorder5andwithabsoluteerrorssummedoverallorderconditionsintheorderof10‐14.Theinfluenceofpopulationsizeprovedtobemorediffusethanforthetarget‐functiontraining.Thismadeitnecessarytohavemoreandlongerruns (up to100,000generations) toachieveconvergence.Furthermore, schemesofhigherorder, having more terms and coefficients, required larger population size (for orderconditiontraining)orinclusionofmoretrainingpoints(fortargetfunctiontraining).Theresearchunderlinestheeffectivenessofthedifferentialevolutionalgorithmasaneasy‐to‐implementheuristicwithfewmodelparameters.TheaccurateresultsoftheRunge‐Kuttaruns showed its persisting capability to avoid local optima, provided that the terminationcriterionisappropriatelytunedsothatrunsdonotendprematurely.
9.4 ReflectionontheworkinChapters8and9
Aswiththeworkofthepreviouschapter,thechoiceforevolutionarycomputingwasinitiallymadewith(anextensionto)GPinmind.PreliminaryGPtestsshowthatevolvingrecurrencerelationsgoesmuchinthesamewayasforstandardSR,butfitnessevaluationbysolvingatargetinitialvalueproblembythecandidatealgorithmsisacomputationalhurdle.Thiscanpossibly be solvedby smart choices in programming implementation. In futurework, testproblems thatvary indifficulty (e.g. stiffness)during the run couldbe consideredornon‐
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error‐based fitness evaluations, such as based on algorithm complexity, or symmetry. Anelegantwayofreducingtherequirednumberofcandidatemodelevaluationsworthtryinghere is the coevolution of fitness predictors (Schmidt and Lipson, 2008). Usingdifferentiationasanexampleofanumericalproblemthatwewanttofindaschemefor,onetrainingpoint consists of twodata sets (a sampled function and correspondingderivativevalues). Applied to this problem, one fitness predictor consists of a set of such ‘trainingpoints’. Findingawell‐trained fitnesspredictor then solves theproblemofwhat functionsarebesttochoosefortrainingthenumericalschemes.Inshort,thinkingaboutthediscoveryofnumericalalgorithmsinthepresentedwayleadstoquestionsabouthowandhowwelltheycanbeoptimised.Canwelookatcertainintegrationalgorithmsaslocaloptimainthesearchspaceofallsolveralgorithms?Euler’smethodistheleastcomplexandfastesttorunbutprovidespooraccuracy,Heunisonestepupfromthis,etcetera.CantheassociatedParetofront(Figure8.16)befoundandwillthisproducemoreefficient solver algorithms?Future general applications includenewnumerical stencils forsolving partial differential equations. Specific examples are algorithms that provide fluid‐solid interface coupling, rules formodel decomposition inmesh‐based CFD,wave‐currentboundaryconditionsandagainturbulencemodelling,tonameafew.
Figure9.10.WeirintheNederrijnatDriel,TheNetherlands(https://beeldbank.rws.nl,Rijkswaterstaat).
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10 Conclusionsandperspectives
10.1 Collectedconclusions
Twolinesofthoughthavebeenpursuedthroughoutthethesis:contributingtosolutionstothe hydraulic engineering problem of flow‐induced gate vibrations and putting newcomputationaltoolstothetest.Indoingso,theworkhasexploredfourdifferentapproaches:physical experimentation, physics‐based numerical modelling, data‐driven modelling forcontrolanddata‐drivensystemidentification.Asurveyofpastresearchshowedthatadampedmass‐springoscillatorpartlysubmergedina flowing fluid provides a useful analogy, but also that the excitationmechanisms can becomplex and the added hydrodynamic coefficients are not readily quantified. It has beenmadeclearthatnumericalmodellingfromfundamentalequationsalonecannotfullypredictthe fluid‐solidbehaviourof real‐life gates.Modelling techniques coupled tomeasuredgateresponsesintroducedinthisstudyareastepforward.ParameterizedmodellingoftheflowaroundthestructurewithafixedgatewascoupledtoaCFD simulation model for resolving the free surface. A benign application of dischargemodelling is achieving gate opening scenarios with smoother discharges guided by PID‐controlled operation. This gave the insight that including near‐field flow modelling instructure operation will enable more sophisticated water reservoir management withrespecttoissuessuchassaltwaterintrusionandfishmigration.Themodelresultscanalsohelptoavertunsafegateuseinextremeeventsandpreventinstabilityofbedprotection.Analysisofdata fromaphysical scalemodelexperimentshowedhowgatevibrationsvaryover a range of conditions signified by the gate opening and the reduced flow velocityparameter Vr. For a rectangular‐bottom vertical‐lift gate with submerged underflow,significantcross‐flowvibrationswereobserved in twodistinctVr regionsatgateopeningsbetweenhalfandfullgatethickness.Mostsignificantly,testsofanadaptedgateprofilewithadded ventilation slots indicated reduced dynamic response forces compared to thestandardprofile.With the frameworkofapplication‐centredmodelling inmind, a finite elementmodelwasset up to simulate turbulent gate flow and the vertically moving gate. The arbitraryLagrangian‐Eulerianmethodwasusedtocreateamovingmesh followingthegatemotion,but it could not be applied to simultaneously compute the free surface disruptions. Thesimulations successfully captured the response of both gate types at Vr ≈ 10. A motion‐induced excitation was observed for the standard rectangular profile powered by wakeentrainment.Theleakageflowthroughtheventilatedslotgreatlyattenuatedthismechanismforthenewgatetype–causingapositivelydampedresponseinstead.Theexperimentandthe simulations combinedgive confidence in the intendedworkingof thenovel ventilatedgatedesign.Furtherstudiesshoulddeterminehowthedesignoftheventilationslotscanbeadjustedforoptimaleffectandiftheattenuationalsoworksforin‐flowvibrationsandothergate types. It is also concluded that at present this type of fluid‐structure interaction
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simulationsistoocostlytocoverawideenoughrangeofconditionsnecessaryforacompleteviewonthegateresponse.Inanefforttooutlineafeasibledata‐drivensystemforcontrollinggatedynamicsbasedonsensordata,itwasfoundthatamachinelearningapproachcanbeappliedtoclassifyfuturestates and hence avoid critical flow‐induced vibrations. Actual implementation of a smartmonitoringsystemforgatedynamicswilldependmoreonrecognitionof itsusefulnessbyflooddefenceauthoritiesthanontechnicalhurdles.A new computational method applied to system identification of dynamical systems,evolutionarycomputinghasprovenitsvalueforinferringmotionequationsfromtimeseries.Several tests on synthetic displacement data indicated that this method is suitable forrecognising self‐excited vibrations, including non‐linear terms. The differential evolutionalgorithmisarobustwaytoderivetheODE’scoefficientswithalimitednumberofstrategyparameters. The computation time of this method is relatively long, but its flexibilityprovides unique possibilities for analysing the vibration system and testing theoreticalconjectures. It has been shown that by including a minimum of system knowledge, it ispossible toderivephysicalcoefficientsof themotionequation fromthemeasuredsupportforcesignal.Finally,ithasbeenshownhowthesameevolutionarycomputingmethodcansolveaspecialreverseinitialvalueproblem,inwhichtheODEandthetimeseriesaregivenandasuitablesolver algorithm is derived. The relative ease and accuracy of the performed numericalexperiments suggest a wider applicability. This raises fundamental questions about thedevelopment and use of numerical algorithms and the customisation of computationalmethods.AllresearchaimslistedinSection1.3havebeenachieved.ThenextsectionproceedswithareflectionontheaccomplishedworkandSection10.3endswithrecommendationsforfutureresearch.
10.2 Reflection
Anevaluationoftheresultsandtheeffortthatwasdonetoarriveatthoseresultsleadstoafewremarks.Inthephysicalexperiment,thechoiceofonlymeasuringtheresponseforceshadaneffectonalllaterinvestigations.Theuseofpressuresensorsinthegateprofilewouldhaveaddedtoan even greater complexity of the experimental set‐up, but would have enabled directexcitation analysis, additional validationof the simulationmodel and input‐output systemidentification.Of course, these pressures are not easilymonitored in prototype structureseither,socomputationsshouldnotbedesignedtorelyonitsavailability.Just as time consuming as physical modelling, CFD simulations are indispensible for FIVresearch – as physics‐basedmodelling generallywill be for thewhole of engineering andscience (NSF, 2006). Persisting points of attention, though, are pinpointing themodellingrequirements and validation. Itwas not producing a response signal, but reproducing the
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exact conditions (water levels, discharge, natural gate frequency) that was the biggestchallenge of the finite element modelling in this study. Had there not been experimentalvalidationdataavailable, thenthesimulationswouldhavetakena fractionof thetimeandstillwouldhaveproducedcolourfulplots–themaincaveatofCFD.A tempting idea is toapplymachine learning to trainablack‐boxmeta‐model thatmimicsthe outcome of the slow and complex CFDmodel. This would then directly give the gateresponse froma set ofwater levels.However, themeaningful subset of simulation resultscoveredaquitelimitedrangeofinputconditions–thereissimplynotenoughtrainingdata.Incidentally,makingextrapolationsandeveninterpolationsfromtheseFSIsimulationsisnotrecommended.Thedata‐drivenmodelproposedinChapter7ismoresuitedtothis,butthecautionwith thismodel is that itsmain goal is to assist in gate operation, i.e. to optimisecontrol,andnottoanalysetheprocess.Computations based on analytical formulae for estimating isolated quantities such ashydraulicstiffness(seeSection2.4andChapters5and6),thatdescribespecificpartsofthefluid‐solid system, have not become obsolete. These can now serve as initial estimates oroptimisation constraints for new system identification methods which seek to derive adescriptionoftheentiresystem.
10.3 Perspectivesandrecommendations
This thesis can be seen as an attempt to pivot solution strategies for vibration issues ofhydraulic structures: continuous monitoring in combination with data‐driven modellingyields valuable information on safety and provides better operational control. As alreadyremarked by Kolkman and Jongeling (1996), hydraulic structures are increasingly beingoperatedremotely.Ifthereducedhumanobservationsofthestructure’sbehaviourthatisaresultofthis,isnotcompensatedbyautomatedobservationtechniques,thenthiscouldleadto higher risks of gate vibrations. Investments of permanent sensor installations on civilengineering structures most probably outweigh current approaches of last‐minutelaboratory testing and occasional fieldmeasurements. Moreover, this idea fits in a widertrendofsmartmonitoringofcivilengineeringstructures(Owenetal.,2001;Magelhãesetal.,2008;Pyaytetal.,2014).Thisnewperspective could contribute to amore comprehensiveproblemsolving strategy(in linewithSection2.5) inthefollowingway.Foranexistingstructure,theideaistofirstgatherdatafromsensorsandthusderivethedominanteigenmodes.Then,thisinformationcanbeusedwithinanautomatedoperationalsystemforavoidanceofcriticalconditionsbymeansofmachinelearning.Thesamedatacouldalsobeusedtoestablishlinksbetweenthegate response and hypothesized excitation mechanisms (in the form of ODEs) withevolutionarycomputing,sothatthecausesofthevibrationscanbeuncoveredandperhapsbeunderstoodbetter.Itisgoodtomentionthatnoneoftheapproachespretendstobeabletopredictwhethersignificantvibrationswillindeedoccurinanewlybuiltstructureornewgatedesign.But thedevelopmentofnewmethods is thought to result in ahigher stateofpreparednessforthepossibilityofunanticipatedvibrationsandamoreeffectivepreventionofcriticalsituations.Allthisdoesnotimply,bytheway,thatsolidknowledgeofthephysicalphenomenaisnotnecessaryforstudyingFIVandforsolvingFIVproblemsinpractice.
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Physicalexperimentsofhydraulicgatesinthelaboratoryhavenotlosttheirvalue–onthecontrary, this remains an indispensable way of performing physical analysis of new gatetypesandtestingreal‐lifedesigns.Additionally,itfulfilsarenewedroleinthedevelopmentofdata‐driven tools.Themainchallenges in this fieldarecontinuity inexpertiseand long‐termreservationofresearchbudget.Insimulationfromfundamentalprinciples(CFD),turbulenceremainsahugechallengeintheforeseeable future. In the specific context of gate vibrations, turbulent flow has to becombined with resolution of the free surface and a moving object. Main concerns arevalidation(withafocusondamping)andexpandingtherangeofphysicalconditionsthatthenumericalmodelcanhandle.Progresswillonlybepossiblewhentheapplicationanddomainknowledgehaveprominentplacesnexttothedevelopmentofcomputationaltechniques.TheframeworkofSection2.3servesasaguideinthis.The data analysis revolution in automated scientific knowledge discovery ignited byevolutionary computing deservedly achieves ample attention. It is expected that moreapplication‐oriented software tools will emerge as the universal success of geneticprogramming is translated to specific engineering disciplines. The full potential for non‐linear system identification will be reached when evolutionary algorithms are combinedmorecloselywithexistingsignalprocessinganddataregularizationtechniques.Theviabilityof automated derivation of solver algorithms is uncertain; the next step is to apply thepresentedmethodtootheralgorithmdesignproblems.
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Summary
The dynamic response of gates of hydraulic structures caused by passing flow poses apotential threat to flood defence safety. Complex interactions between the turbulent flowandthesuspendedgatebodymayinduceundesiredvibrations,aprocessthatisnotreadilyassessedorcapturedindesignrules.Thisthesiscontributestoabetterunderstandingandpreventionof the gate vibration problemby building on previous research andproposingnewsolutionapproaches.Aphysical scalemodelexperimentwasconducted to investigatecross‐flowvibrationsofavertical‐lift gate with underflow. This was done for two profiles: a standard rectangularprofileandanadaptedprofilewithventilationslotsadded.Undersubmergedconditionsandat small openings, the ventilated gate showed significantly reduced vibration amplitudes,thisisshownmostprofoundlyforthedimensionlessreducedvelocityVrintherange2<Vr<3.5. The data obtained from the experiment were used for validation of physics‐basednumerical simulations with the finite‐element method. An arbitrary Lagrangian‐Eulerianmesh enabled a transient analysis of the local flow field, boundary pressures and theresultingmotion.Thesimulationsgiveinsightintohowtheleakageflowoftheadaptedgatetypediminishes theobservedmovement‐inducedexcitationof the flat‐bottomgate atVr ≈10: by suppression of entrainment from thewakewhich is held responsible for pressurefluctuationsclosetothetrailingedge.Withthemovingmeshapplied inadifferentway,numericalsimulationsweremadeofthefreesurface.Itisshownhowthesenon‐hydrostaticcomputationsoftheflowaroundafixedgatecanaidinthepredictionofgatedischarges–somethingthatlargersystem‐scalemodelsoften omit. In particular, numerical tests illustrate how this may support operationaldecisions of gate opening and closure, optionally using automatic control, such thatdischarges and local flow velocities have minimal detrimental impact regarding scour,ecologyandvibrations.Theinstallationofsensorsonhydraulicstructuregatesconstitutesapromisingnewwayofavoiding critical vibrations, as introduced in this work. A data‐driven system is outlinedwhere continuous monitoring of accelerations fills a database. Machine learning is thenappliedtoclassifytheobservedamplitudesasafunctionofgateopeninganddimensionlessreduced velocity, thus training the system for evaluation of future states. The proposedcontrolsystemisaimedatatimelyrecognitionoftheriskofflow‐inducedvibrations,sothatsuitableoperationalmeasurescanbetaken.In addition to the use of the measured vibration data for validation of a physics‐basedsimulationmodelandfordemonstrationofcontroloptions,athirduseisexploredaswell.Evolutionary computing comprises a class of heuristic computational methods that hasrecently started to reveal its huge range of possibilities. In this thesis, the differentialevolutionalgorithmisusedtoidentifythecoefficientsofsecond‐orderdifferentialequationsfromtimesignals.Ofspecialinterestistheidentificationofself‐excitedoscillations,themost
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harmful type forgates inhydraulic structures,which typically containnon‐lineardampingterms.Numericalexperimentsoftheproposedmethodassumeequationstructuresandusethe velocityonly at the first time step. It is furthermore shownhow the technique canbeextended to recover the motion equation from only the support force signal. Intendedapplications are pinpointing self‐excited vibrations for early‐warning purposes and theanalysisofintractablenon‐linearsystemsingeneral.In themore abstract final chapter, the same differential evolution algorithm is applied toderive numerical solver schemes for differentiation and integration problemswith knownsolutions.Thecoefficientsoffinitedifferenceformulaeandanumberofsingleandmulti‐stepintegrationmethodsaregenerated.Mostinterestingly,newfifth‐orderRunge‐Kuttaschemeswere reverse engineered with remarkable accuracy using the set of order conditionequations for fitness evaluation. This part of the study could be valuable for futuredevelopmentsinthedesignofadvancednumericalmethods.
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Samenvatting
Dedynamischeresponsievanschuiven inwaterbouwkundigeconstructies tengevolgevande passerende stroming vormt een mogelijke bedreiging voor de waterveiligheid. Decomplexe interacties tussen de turbulente stroming en het schuiflichaam kunnenongewenste trillingen induceren,dit iseenprocesdatnietgemakkelijk in teschatten isenniet zondermeer te vatten is in ontwerpregels. Deze dissertatie draagt bij aan een beterbegrip en voorkoming van schuiftrillingsproblemen door voort te bouwen op voorgaandonderzoekendoornieuweoplossingsrichtingenaantedragen.Een fysisch schaalmodelexperiment is uitgevoerd om trillingen dwars op de stroming vaneenverticaalbewegendeschuifmetonderstromingteonderzoeken.Ditisgedaanvoortweeschuifprofielen: een standaard rechthoekig profiel en een aangepast profiel metventilatiesleuven in de onderrand. De geventileerde schuif ondervond significantverminderde trillingsamplituden in verdronken toestand bij kleine schuifopeningen, dit ismetnameaangetoondvoordedimensielozegereduceerdesnelheidVrinhetgebied2<Vr<3.5. De data verkregen met dit experiment zijn gebruikt voor validatie van numeriekesimulaties gebaseerd op fysische beginselen met de eindige elementen methode. Een‘arbitrair Lagrange‐Euler’‐rooster maakte een tijdsafhankelijke analyse mogelijk van hetlokale stromingsveld, de drukken op de schuifrand en de resulterende beweging. Desimulaties geven inzicht hoe de lekstroom van de aangepaste schuif de waargenomenbewegingsgeïnduceerdeexcitatieverminderenvaneenschuifmetplatteonderrandbijVr≈10; te weten door onderdrukking van impulsinmenging vanuit de neer, hetgeenverantwoordelijkwordtgeachtvoordrukfluctuatiesnabijdebenedenstroomserand.Numerieke simulaties van het vrije wateroppervlak zijn gemaakt door het bewegenderekenrooster op een andere manier toe te passen. Er wordt aangetoond hoe aldus niet‐hydrostatische berekeningen van de stroming rondom een vaste schuif de schuifafvoerkunnenvoorspellen–watgrootschaligesysteemmodellenvaaknalaten.Denumerieketestenillustreren in het bijzonder hoe operationele beslissingen voor sluiten en openenonderbouwdkunnenworden,eventueelmetbehulpvaneengeautomatiseerdregelsysteem,opdatafvoerenenstroomsnelhedenminimaalnadeligegevolgenhebbenvoorlokaleerosie,ecologieentrillingen.De installatie van sensoren op schuiven van waterbouwkundige constructies vormt eenkansrijke manier om kritische trillingen te voorkomen, dit idee wordt in dit werkgeïntroduceerd. Een datagestuurd (‘data‐driven’) systeem wordt beschreven waarin eendatabank wordt gevoed door continu de schuifversnellingen te meten. Vervolgens wordtmachinaal leren toegepastomdewaargenomenamplitudes te classificerenals functievanschuifopeningendimensielozegereduceerdesnelheid;zodoendetrainthetsysteemzichzelfvoorevaluatievantoekomstigetoestanden.Hetvoorgestelderegelsysteemisgerichtopeentijdige herkenning van risico’s op stromingsgeïnduceerde trillingen, zodat geschikteoperationeleingrepenmogelijkzijn.
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Naast het gebruik van trillingsmetingen voor validatie van een numeriek simulatiemodelgebaseerd op fundamentele vergelijkingen en voor het demonstreren van regel‐mogelijkheden, is nog een derde gebruik verkend. Evolutionaire algoritmen vormen eengroep heuristische rekentechnieken die recentelijk hun enorme reikwijdte zijn begonnenprijstegeven.Indezedissertatiewordthet‘differentiëleevolutie’‐algoritmegebruiktomdecoëfficiënten van tweede‐orde differentiaalvergelijkingen van tijdsignalen te identificeren.Vanspeciaalbelangzijntrillingendoorzelf‐excitatie,diezeergevaarlijkzijnvoorschuivenen normaliter gepaard gaan met niet‐lineaire dempingstermen. Numerieke experimentenvandevoorgesteldemethodegaanuitvanvastevergelijkingsstructurenenhebbenenkeldesnelheid op de eerste tijdstap nodig. Bovendien wordt beschreven hoe de techniek kanwordenuitgebreidomdebewegingsvergelijkingteherleidenaandehandvaneentijdsignaalvandeophangingskracht.Beoogdetoepassingenzijnhetherkennenvanzelf‐excitatievoorgebruikinwaarschuwingssystemenenanalysevanniet‐lineairesystemeninhetalgemeen.Inhet abstractere laatstehoofdstukwordthetzelfde evolutionaire algoritme toegepast omnumerieke rekenalgoritmes af te leiden voor differentiaalproblemen met bekendeoplossingen. De coëfficiënten van rekenmethoden voor eindige differenties en numeriekeintegratie zijn automatisch gegenereerd. Hetmeest interessante resultaat behelst ‘reverseengineering’ van nieuwe vijfde‐orde Runge‐Kuttaschema’s met opmerkelijkenauwkeurigheid, gebruikmakend van de ordeconditievergelijkingen voor evaluatie van de‘fitness’.Ditonderdeelvandestudiekanvanwaardezijnvoortoekomstigeontwikkelingeninhetontwerpenvangeavanceerdenumeriekemethoden.
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Postscript
BreakingwiththetraditionofendingthethesiswithasentimentalreviewofthePhDasanirreversiblecoming‐of‐agetypeofjourney,Iwillgiveashortoverviewoftheeventsthatledme to starting a PhD, some subjective reflections on the work and a few statements ofgratitude.MyfirsttasteofsciencewasduringmyMasterprojectatDelftUniversity–afieldexperimentintheriverSpreenearBerlin.AftergraduationIworkedfortwoandahalfyearsatDeltaresinDelft,anon‐profitwaterinstituteforresearchandengineeringconsultancy.Thereitwaspossibletoaddlaboratoryexperimentsandnumericalmodelling(CFD)tomyfluiddynamicsportfolio,andIstartedworkingonweirsandnavigationlocks.ThegatevibrationexperimentinApril 2011gaveme the idea that itwouldbenice tohavemore time for research.TheprogresstowardsfindingasuitablePhDprojectwas initiallyslow;until IreadaboutPeterSloots project in Russia and got in contact with him. Combining a topic from hydraulicengineering from first‐hand practice with computational science methods, some of themcompletelynewtobe,turnedouttobeasgreataplanasIhadenvisioned.Theadventurousextradimensionof coursecame fromhavingSaintPetersburgasmyhome town for threeyears.ThisfulfilledthewishofspendingmoretimeinthisgreatcitywhichIhadvisitedonashortholidaytwoyearsearlier.Letme share a few thoughts on the researchwork. Forme it has always been clear thatthings becomemore interesting and valuablewhen different disciplinesmeet. Ideas fromseparateworldsthataretrivialintheiroriginalcontextscanunexpectedlybecomepregnantwith new meaning and impact. This is how creativity is awarded. Sometimes, though, itseems that with it comes a curse of multi‐disciplinarity: in evaluating multi‐disciplinaryresearch, experts from different fields each isolate and critically assess one aspect of theworkandthenconcludethatitdoesnotcontributemuchtotheirparticularfield.Instead,itwouldbebettertojudgethecombinedworkasawholebylookingattheoveralladdedvalueandthevalidityof theapplication.At theendof theday, the twodriving forcesofscience,rigourandoriginality,needtobebalancedsomehow.Butoriginalitythrivesbestinfarawaywastelands, while rigour is rooted in highly cultivated grounds. The challenge for thescientist and especially for the budget managers and policy makers is to recognise andrespectthis.Research,includingPhDprojects,canonlybesuccessfulif,ontheplannedroutetoX, it is allowed to try a sidepathY and there is time to get lost in adarkalleyZ.Oncearrivedand settled inX, lookingback on the journey, it often turns out that a small thingfoundhalfwaybetweenYandZisactuallythemostvaluableresult.Thisiscapturedbythenice‐soundingtermcollateralknowledge.Forme, theadditionofevolutionarycomputingbroughtnewenergy to theprojectaroundthe notorious halfway point – I thank Peter for bringing this topic to my attention. Asubstantialamountofworkwentintogeneticprogramming,andalthoughthisdidnotmakeitintothethesis,itwasgreatfunandsparkednewideas.WhatIfoundwasthatadangerinthecomputationalsciencesistodeviseanewmethod(algorithm),presentitwithouttesting
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it to different problems andwithout comparing it to existingmethods and forget to givefeedbacktotheapplicationfield(asalsomentionedbyWagstaff,2012).Ontheotherhand,adanger in (hydraulic) engineering is to solve real‐world problems exclusively fromexperienceandfailtomakeconnectionswithuniversaltheories.Thecuretobothdangersisgenerally to lookbeyondthedetailedactivitythat iscentral toyourwork,realise that it isgoodtozoomoutregularlyandtrainyourselfinswitchingbetweenaconcreteandabstractfocus.Comparing the validity and accuracy of physical experiments (in the lab or in situ) withnumerical experiments (in silico) is a common activity that can easily give endless clichédiscussions.Researchers typically choose sides early in life andarenot evenawareof theprejudices they hold against the opposite paradigm. In the context of this thesis it hardlyrequiresexplanationwhyitisuselesstospeakaboutphysicalandnumericalmodellingasifone would be better, more accurate or cheaper than the other. A possible way to unitequality control of all types of experiments is to think of validation as a process of criticalfeedback in which data hold a central place – compare Section 3.2. The value of suchframeworks is to support and motivate choices when setting up new models and modelapplications.However,usingtoogeneralrulesandphilosophieswillmakeitmeaninglessinterms of practical use. Finally, I would like to point out two phenomena from personalexperience that always set off warning bells. The first is reasoning against nature, thishappenswhensomeonesays thatmeasurementsnicely lineupwithnumericalpredictionsfromtheoryorthatthemeasurementsare inaccuratebecausetheyaretoofaroff. It isnotnaturethatdoesordoesn’tcomplywiththeory,butthetheorythatdescribespartsofnaturewellornotsowellundercertainconditions.Itcanbethecasethatthoseconditionsarenoteasily matched by the (physical or numerical) experiment, but this is not nature’s fault.There are many subtle variations on this. The second phenomenon is the creation ofcomputation‐only realities, this occurswhen initial numerical results show such appealingpropertiesthatthecomputationsareproceeded,expandedandtrustedwithoutcheckingtheoverlapwiththe(physical,biological,etc.)realmthattheytrytorepresent.Inextremecases,theseendeavours createawholeworldofbeautyandelegance,which canbe insightful inmany different ways, but these ought not to lure us into thinking that the results havecomparablemeaningsobservableoutsideofthecomputationalrealm.Now it’s time to thank all the people who helped me, starting with my parents whosupportedmeineverypossibleway;especiallythesmoothtransitionsbetweenRussiaandTheNetherlandswouldhaveneverbeenpossiblewithoutyou.IoweagreatthanksofcoursetoPeterSloot,withoutwhomIwouldprobablystillbewritingresearchproposalsforDutchresearch funds. You made it possible for me to take an inspirational free dive intocomputational science and exchange the predictable world of planning, preparations anddiscussionsforanunforgettableRussianlife.NextupisofcourseValeria,youwereagreatand tireless supervisor for me, who always had swift replies and practical solutions thatresultedinmakingalldeadlines.IalsowanttosayabigthankstoDeltares,andinparticulartheHarbourCoastalOffshoregroupofKlaasJanBos,forsupportingmethroughoutthePhDperiod.Iamgratefulthatthelifelinealwaysremainedintact,eventhoughIvisitedDelftonlya handful of times. Hopefully the work has relevance for future projects and appliedresearch. A special thanks goes to Tom Jongeling, a rare gate vibrations specialist whoretiredfromworkduringmyPhD.Althoughwedidnothavecontactafter I leftDelft, Iam
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sure you will find in this work familiar basic ideas and principles that you taught me.Unknowingly at the time, experiences from projects that we work on together becamecornerstonesofsomeofmypapersandchaptersinthisthesis.AtITMOUniversitymythanksfirstofallgoouttoAlexanderValerevichforallowingmeintohis department as the first foreigner, giving me the freedom to do my researchindependentlyandgivingmethetrusttocontinuemyworkallthewayuntilthedefence.Ihad a good and productive time at НИИ НКТ. My teaching experience and our trips toconferencesinAmsterdamandAustraliawereallquitespecial; it feltstrangetoleaveafterthree years. Of course I thank Vitya, for surviving all this time on the same four squaremeters ofwindowless office spacewithme,where no oxygenmoleculeswere allowed in,only liters of instant coffee.Our commondedication to anti‐mainstreammusic resulted inniceplaylists,TheCaretakerandAutechreendedupasmyall‐timefavourites.Anna,thankyou for the good times, especially the epic New Year and skiing trip to Siberia which isimpossibletoeverforget.AlsoIhavegoodmemoriesofthetimeLouiswasworkingatITMO,yourenthusiasmforscienceworksverycontagiousandIwishyougoodluckwithyourPhD(notthatyouneedit).Inshort,IwillcherishgoodmemoriesoftheentireITMO‘kollektiv’.Ябудускучатьповсемвам!Ispentthesummerof2013attheScienceParkoftheUniversityofAmsterdam,whereIhadanicetimeintheComputationalScience(SCS)group:Paula,Rick,Kees,Mike,Guusje,Brechtandtherest,Iwishyouallthebest!ThankstoBibiandTrudieforbeingwell‐behavedhousemates in the capital’s presidential area during that summer. Another Amsterdam citizenwhomIshallnotforgettothankisJacques(Sjaak),intheendyouconvincedmetodoaPhD.Inmy home town of TheHague I owemore than they realise to Floris (Frits) andHJ forlettingmedumpapileofboxesintoyourhomes,“Ipromisetopickthemupreallysoon”.ReturningagaintoRussia,IwanttothankAndreifortheflexibilityinhousingmerightinthecenterofthecityforquitesometime.ThentherearetoomanydearfriendsImetinPiter,anextremely incomplete shortlist: Kostya, Dasha (2x), Ksyusha, Marina, Rita & Dima, Guzel,Julia,Johan&Ksenia,Lodewijk,Sasha‘fromRussia’,Lily,GiaandDiana–Iplantomeetallofyou in the future. I alsowant to congratulate Natalia and Alexanderwith their PhD, welldone!IfinishbythankingLucasforhishelponthecoverdesign.AndlastbutcertainlynotleastIthankLouisandAnnebethforbeingmyparanymphs!
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Curriculumvitae
ChristiaanErdbrink,[email protected]–2007 DelftUniversityofTechnology:CivilEngineering MScwithhonoursinEnvironmentalFluidMechanics
2000–2002 UniversityofAmsterdam:MathematicsandStatistics completedmostfirstandsecondyearcourses
1998–1999 TheUniversityofEdinburgh:EngineeringandMathematics WinnerofHorsburghprizeformathematics
1992–1998 GymnasiumHaganum,TheHague:pre‐universityeducationEmployment2011–2014 UniversityofAmsterdamandSaintPetersburgNationalResearchUniversity
ofInformationTechnologies,MechanicsandOpticsPhDstudentComputationalScience
2007–2011 DeltaresAdviser/researcheratdepartmentofRiverEngineeringuntil2010,afterthatdepartmentofHarbour,CoastalandOffshoreEngineering
Experience2006 Leibniz‐InstitutfürGewässerökologieundBinnenfischerei(IGB)inBerlin FieldmeasurementcampaigninSpreeRiveraspartofMSc
2005 DredgingcontractorVanOordNVinDubai,U.A.E. Internship.StudyonstabilityofreclamationsatTheWorldproject
2005 BlackSeaCoastalAssociation,Bulgaria. CoastalmorphologystudyforbeachenhancementinVarnaAspartofthePhD,Ipresentedmyworkatseveralscientificconferences,taughtEnglishandevolutionary computing and reviewed papers for scientific journals. At Deltares, formerlyWL|Delft Hydraulics, I worked as an engineering consultant and researcher on numeroushydraulicengineeringprojects(alsoasprojectleader),amongwhichthedesignofthethirdset of locks in the Panama Canal and I co‐organised the 2008 NCR conference for riverstudies. As a student in Delft, I was active at the Delftsch Studenten Corps in variouspositionsforwhichIreceivedgrants.Also,between2002and2006Iworkedasaguideatthe Rijkswaterstaat information center at the Maeslant storm surge barrier. Languages:Dutch(mothertongue),English,German,Russian.