Numerical Study of Dam-Break Flow OMID SEYEDASHRAF 1, ALI AKBAR
AKHTARI 2, HEJAR JEBELI AQDAM 3 M.Sc. Student of Hydraulic
Engineering 1, Assistant Professor 2, M.Sc. Student of Hydraulic
Structures 3 [email protected] [email protected]
[email protected]
AbstractWhenadamisbreached,calamitousflashfloodingoccursastheimpoundedwater
flees through the opening into the downstream river, which is
followed by massive live and property lost. However, taking into
account the large scale of the phenomenon, a
numericalschemeremainsthe
preferredmethodfortheprovisionalstudies.Finite Volume Method (FVM),
Finite Difference Method (FDM) and Finite Element Method (FEM) are
the threemain numerical methods used in simulating hydraulic
incidents. The use of non-linear Shallow Water Equations (SWEs) has
become standard for
dam-breakflowmodelingandhasbeenprovedbothusefulandconvenient,albeitthe
existenceofsignificantambiguitiesduetothesimplifyingassumptionsand
approximationsusedintheSWEs.Inthepresentstudyattemptsweremadeto
numericallyinvestigatethefluidflowcharacteristicsofdam-breakphenomenon
throughdifferentgoverningequationsandnumericalmethods.Theresolutionof2D
SWE and 3D Navier-Stokes (NS) equations in the dam-break
simulation, is put side by side the experimental data, validated
and compared the accuracy of both models. The
CFDanalysisisexecutedbymeansofthecommercialsoftwarepackageFluent.The
volume of fluid (VOF) model is employed to depict the air-water
interaction at the free surface. It has been concluded that, both
2D SWE and 3D NS models arecapable of
capturingthedam-breakshocksreasonablyaccurate,yet,consideringtheCPUtime
andoverallevaluations,the2DmodelwiththeSWEasthegoverningequations,is
preferable to the 3D NS model in numerical simulations of the
dam-break phenomena. Keywords: dam-break flow, shallow water
equations, navier-stokes equations, open channel bend,
computational fluid dynamics.
2Introduction Dam-break flow is the immediate release of
initially stationary water body by removing a vertical obstacle,
such as in case of a reservoir or a dam failure, the after effects
transient flow over the bed is termed as dam-break flow. When a dam
is breached, calamitous flash
floodingoccursastheimpoundedwaterfleesthroughtheopeningintothedownstream
river.Generally,theresponsetimeavailableforwarningismuchshorterthanthatfor
acceleration of runoff floods, called shocks. Consequently, the
possibility of loss of lives is much deplorable. The emphasis on
dam failure hazards have become so well known that it
isprotectedbytherulesof InternationalHumanitarianLaw
(IHL)anddamsshallnotbe
madetheobjectofattackduringarmedquarrels,ifthatmaycausebrutallossesamongst
the civilian populations. There are 17 critical infrastructure
areas like national monuments, energy, chemical and agricultural
systems, originally listed in the USA, Homeland Security
PresidentialDirective(HSPD)7of5/7/2007obligationtoensuretheprotectionof
resourcesincludingdams.However,therehavebeenaround 200importantdam
and reservoir failuresin theworldsofar inthe
20thcentury.Theworldsmostawfuldam
failureoccurredinHenan,aprovinceinChina,inAugust1975,whentheBanqiaoDam
andtheShimantanDamfailedhorrendouslyduetotheovertoppingcausedbytorrential
rains.Inthisincidentaround85,000peoplediedbecauseofthefloodingandmanymore
losttheirlivesduringitsconsequentplagueandstarvationandmillionsofoccupantslost
their houses [1]. This disastrous is analogous to the events of
Chernobyl and Bhopal for the
nuclearandchemicalindustries[2].Figure1showstheten-yearrunningaveragenumber
ofdamfailuresasavailableinthedigitallibraryofNationalPerformanceofDams
Program(NPDP)[3].The running average is shown forthe complete
historic record (all
failures)andfortherecordexcludingthemanysmalldamsthatfailedduringthe1994
Georgia floods. Figure1:Ten-year running average number of dam
failures (excluding the many small dams that failed during the
1994Georgia floods)
Incomparingwithexperimentalinvestigationsandbecauseofthescaleoftheincident,
numericalmethodscouldbemoreattention-gettingtopredicttheflowbehaviors,
hydrographsandroutings.Numericalstudiesofcurrentsinchannelsandrivers,mainly
drivenbyfloods,haveaninterestingrangeofapplicationsinenvironmentalhydraulics,
navigationandprovidingsafedrinking water tothe rural
massesandindustryuse.However,thedynamicsofthedam-breakwavepropagationisrathercomplexandits
behaviordonotmeetthetermsoftheregularassumptionsofconventionalsteadyand
gradually varied open-channel flows.
3Asarecentinvestigation,SoaresFrazaoetal.conductedaseriesofnumericalmodeling
and laboratory experiments of dam-break phenomenon on a 90o bend of
rectangular cross
sectionwithastraightoutletreach[4].Theysolved1Dand2DShallowWaterequations
(SWEs) in finite volume method and compared to the taken pictures
of the water flow. He
measuredbothvelocitiesinthebendandwaterdepthprofilealongthechannels,noticed
thatthe2Dmodelwasinagoodagreementwiththeexperimentaldata.Yingetal.have
also developed numerical models for flows generated by a dam
failure or levee breaching process using a conservative form of
SWEs, for more information refer to [5].
NumericalmodelingofsuchphenomenausingphysicalillustrationssuchastheNavier-Stokes(NS)equationscanfrequentlybeawkward,duetotheextentofthemodeling
geometries as well as through resolving free surfaces. SWEs, of
which there are a number of demonstrations, provide an easier
picture of such
phenomena.DuetoComplexitiesofthenonlinearterms,computationaldomainandbottom
topography,numericalapproximationsaretobeexpected.Ofthese,finitedifference
methods(FDMs),finitevolumemethods(FVMs)andfiniteelementmethods(FEMs)are
the most common. Normally, FEMs provide a better resolution of the
flow domain but due
totheirlargematrixsystems,aremoreexpensivetoimplementandtoexecute,unless
modern methods e.g. MultiFrontal, are employed to solve the huge
linear system [6]. In the numerical studies, since spurious
oscillation may occur near the discontinuities, in which it is
inevitable in a numerical dam-break model, and can substantially
corrupt the results and generateinstabilities,ithasbeendifficult
togivesuch transcriticalflowsaclear interpretation. Accordingly,
the results obtained from nearly all classic numerical methods
sufferfrom
thesenumericalfluctuations.Toovercomethis,avarietyofhigh-orderwell-balanced
schemes for SWEs are developed and can be found in the literature
(e.g. Toro E.F [7], LeVeque [8], Kurganov and Levy [9]) these
schemes construct excellent estimates of the quasi-steady solutions
and non-stationary steady
states.Inthepresentstudy,attemptsweremadetonumericallyinvestigatethefluidflow
characteristicsofdam-breakphenomenonthroughdifferentgoverningequationsand
methodologies.Theresolutionofthe3DNSequationsinFVMmodelisputsidebyside
the data obtained from a 2D SWE model and the experimental data by
Soares Frazao et al.
[4].Toresolutethegoverningequations,thecommerciallyavailableCFDpackage
FLUENT was utilized. Results were validated and compared to the
ones obtained from the 2D model. 3D Reynold-Averaged Navier
-StokesThegoverningequationforCFDisbasedonconservationofmass,momentum,and
energy.TheusedCFDpackagemakesuseofaFVMtosolvetheNSequations.
Advantagesofusingthesetypesofcomputationaltoolsarethepossibilityofevadingthe
SWEs restrictions, allowing a vertical description of the different
variables of the flow e.g.
thevelocityprofileandtheabilitytomodelturbulentcurrentsbyvariousmethodologies
suchasthek-model.MakinguseofaVolume-of-Fluid(VOF)approach,wewouldbe
abletoresolvethefreesurfaceevolution.FVMinvolvesdiscretizationandintegrationof
the governing equation over the control volumes. The fundamental
equations for transient state laminar current are conservation of
mass and momentum. In this research as the heat transfer or
compressibility is not involved, the energy equation is deleted
from the equation system. The governing equations are stated
as:
4( ) ( )miiSxut=cc+cc ( )( ) ( )jj iijjij i jj iixu uxuxux xpxu
utuc' ' c+(((
cc+cccc+cc =cc+cc (1) (2) where t is time, ui is the i-th
component of the Reynolds-averaged-velocity, xi the i-th axis
(withtheaxisx3verticalandorientedupward),
isthewaterdensity,pistheReynolds
averagedpressure,gistheaccelerationduetothegravity,
istheviscositywhichis equal to zero in this study and Sm is the
mass exchange between the two phases. It should
benotedthattheunsteadysolverwillbeusedtogetthevelocitiesandothersolution
variablesnowrepresenttime-averagedvaluesinsteadofinstantaneousvalues.Theterm
appeared ( )i ju u ' ' is called Reynolds-stress. This term is
obtainable from the Boussinesq hypothesis which links
Reynolds-stresses to the mean rate of deformation. However, in this
work due to the large scale of the phenomenon, this term is
disregarded. 2D Shallow Water Numerical Model
Sincethe17thcentury,Newton(16431727)andLeibniz(16461716)shapedtheworld
ofmodernmathematicsbyintroducingcalculus.Withdeferentialincalculus,people
startedtothinkaboutdeferentialequations.TheSWEsalsocalled
Saint-Venant equations initsone-dimensionalformandafter
AdhmarJeanClaudeBarrdeSaint-Venant, are a system of hyperbolic
partial differential equations that depicts the flow below a
pressure surface in a fluid. The non-linear SWEs are regularly used
for modeling flows in which the depth D is much less than the
wavelength L, like oceanographic or atmospheric fluid flow. Models
of such systems lead to the calculation of areas eventually
affected by
pollution,coastalerosion,andpolarice-capdissolving.ThemostfrequentlyusedSWE
form in dam-break investigations, which is derived from the
Navier-Stokes equations, is as follows: 0t x yU E F S + + + = (3)hU
uhvh ( (=( (
(4)222uhghE u huvh ( ( (= + ( (
(5) 5222vhF uvhghv h ( ( (=( (+ (
(6)0bxxbyyS ghbghbtt ( ( ( (= + ( (+( (7)
ThevectorEandFaretheso-calleduxvectorsandSrepresentsthetopographicaland
frictionalsourceterms,wheregisthegravitationalacceleration.uandvarethedepth-integratedvelocitycomponentsinthexandydirections,respectively,bisthebottom
elevation and h is the water depth. GEOMETRY and MESH GENERATION
Differentgridsizeswereinspectedtoinvestigatethesensitivityandtheprecisionofthe
results. To obtain sound data of the shock propagation and the flow
depth, the channel was
discretizedbydiminutivecells.Around70longitudinal,20latitudinaland22altitudinal
segmentsarecreatedinthespecifiedchannel,asaresultthethree-dimensionalflow
domain was splitted into a total number of 39048 hexahedral
non-overlapping cells which extent from 0.025m to 0.1m. Out of
various possible meshing schemes, the chosen form is suitable for
the accuracy, computational costs and the CPU time of the
convergence. Due
totheeliminationofturbulencetermsinequations(1)and(2)andneglectingthewall
functions,aregularunstructuredgridissufficedtomodelthefluidflow.Figure1and2
respectively show the geometric layout and the plan view of meshing
form. Figure2:Schematic representation of the computational
domain.
6 Figure 3: Meshing form used to perform the computations.
SOLUTION METHODOLOGY
Thegoverningequations(1)and(2)areasetofconvectionequationswithvelocityand
pressurecouplingbasedonthecontrolvolumetechnique.Thegeneralpurposecode
FLUENTwasemployedforallthenumericalsimulationspresentedinthisinvestigation.
The code employs the FVM in conjunction with a coupled technique,
which solves all the
transportequationssimultaneouslyinthewholedomainthroughafalsetime-step
algorithm.ConvectiontermsarediscretizedusingthethirdorderMonotoneUpstream
CenteredSchemeforConservation Law (MUSCL). The linearized system of
equations is
preconditionedinordertoreducealltheeigen-valuestothesameorderofdegree.
Pressure-Implicit with Splitting of Operators (PISO) technique is
engaged to deal with the
problemofvelocityandpressurecoupling.PISOmethodsincorporatepressureimpact
throughmomentumequationsintocontinuityequationtoattaincorrectionequationsfor
pressure.TheVolumeofFluid(VOF)methodwasemployedtosimulatetheair-water
interaction.TheVOFmethodwasdevelopedbyHirtsandNicholas[10]andthe
formulationreliesonthefactthattwoormorephasesarenotinterpenetrating.Foreach
extra phase that added to the system, a variable is introduced in
the volume fraction of the phase in the computational cell. In each
control volume, the volume portions of all phases
sumtounity.Outflowboundaryconditionwaschosenastheoutflowbasinwithtwo
separateoutletswiththesamegroupID.Symmetryboundarycondition,inwhichallthe
normal components and gradients are kept zero, was chosen as the
upper surface boundary.
Thesidesandbottomsurfacesaredefinedaswallboundarycondition.Apressurebased
solver is used to solve the equations since the flow is
incompressible.
RegardingtheCFLcondition(whichisaverylimitingconstraint on
thetimesteptand was named for its originators Courant, Friedrichs,
and Lewy) with a value of 0.25, a set of
5600timestepsof0.0025secondswiththemaximum40iterationspertimestep,was
conductedtosolvethetransientcurrent.Usinganordinaryunstructuredgridhas
considerably enhanced the acceleration of convergence. Calibration
and Validation
AlthoughtheCFDpackageFLUENTiswidelyusedforengineeringapplicationsand
scientificinvestigationsbutvalidationofthenumericalmodelsisalwaysessential.The
7numerical results for the dam-break flow are validated by
comparing thenumerical results with the measurements made at
laboratorial scale by Soares Frazao et al. Figure 2 depicts
thewavefronttrackingofa2DmodelwhileFigure3showsresultsofthecomputed3D
model. Figure 4: Computed (2D Shallow-Water model) positions of
dam-break front, at interval of 0.1 s Figure 5: Computed (3D NS
model) positions of dam-break front, at interval of 0.1 s
Duetoeliminationofthevertical coordinate
fromtheflowequationsinSWEs,The2D
modelcouldntmodelthesecondarycurrents.Consequently,asdepictedintheprevious
figures,discrepanciesinthenumericalresultsbetweenthe3Dand2Dmodelsare
inevitable. In the following figures, water depth profiles are
compared to the experimentally measured ones. Figure 6:
Experimental and computed (2D and 3D numerical models) flow
profiles, t=5(s).
8 Figure 7: Experimental and computed (2D and 3D numerical
models) flow profiles, t=7(s). Conclusions
Theevaluationoftheresultswithexperimentaldatapermitstoillustrateaconclusionon
efficiency of a considered method. Discrepancies have been noted,
between models using different mathematical schemes and equations.
As the gateway is detached instantaneously a shock wave is made and
propagates in through the downstream channel and a reflective
negativewavefrontisgenerated,whichstartstravelingupstreamintothereservoir.Flow
regimetransformsfromsubcriticaltotranscritical,andreachestosupercriticalflowat
various section as the dam-break flow propagates
downstream.Thecomparisonofthewaterdepthprofileoftheexperimentsandthenumericalresults
showagoodmatch.The3D,NSmodelhascapturedtheshocksoutlineprecisely
(particularly at the time step t=7s); however, the 2D, SWE results
seems more realistic and
depictstheshockpropagationproperly.ConcerningFigures6and7,ahydraulicjumpis
noticeableattheinletofthechannelinbothexperimentalandthe3Dmodel,yet,dueto
eliminationofthevertical coordinate
(x3)fromtheflowequationsinSWEsand
consequentlyverticaldescriptionofthedifferentvariablesofflow,thisisomittedinthe
2D model. Also, a temporal fluctuation of water depth at the tip of
the 90o is shown at bothfigures. At this location, there is no
reflective wave front in the 3D model, but progressive
shockfrontsfromtheupstreamsideoriginatetheformationofpeaks.Thesimulated3D
modelproducedagoodmatchwiththephotographedwaterdepthsatthetimestept=7s;
though,themodelunderestimatedlesswaterthantherealitypassesthebendatthetime
step t=5s and this led to lower depth before the bend.
Evenifthewaterdepthistosomeextentunderestimated,nevertheless,thetimeof
appearance of inundation peaks are depicted precisely. Eventually,
it can be concluded that
both2DSWEand3DNSmodelsarecapableofcapturingthedambreakshocks
reasonably well but considering the accuracy and CPU time and over
all evaluations, a 2D
modelwiththeSWEasgoverningequationsismoreappropriatethana3DNSmodelin
numerical simulations of the dam-break phenomena. 9 References [1]
Qing, D. : The River Dragon has come!: Three Gorges dam and the
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