Suspension-Driven gravity surges on horizontal surfaces: Effect of the initial shape

O

pen

A

rchive

T

OULOUSE

A

rchive

O

uverte (

OATAO

)

OATAO is an open access repository that collects the work of Toulouse researchers and

makes it freely available over the web where possible.

This is an author-deposited version published in :

http://oatao.univ-toulouse.fr/

Eprints ID : 17810

To link to this article : DOI:10.1016/j.compfluid.2017.03.016

URL :

https://doi.org/10.1016/j.compfluid.2017.03.016

To cite this version : Zgheib, Nadim and Bonometti, Thomas and

Balachandar, Sivaramakrishnan

Suspension-Driven gravity surges on

horizontal surfaces: Effect of the initial shape.

(2017) Computers and

Fluids, vol. 158. pp. 84-95. ISSN 0045-7930 Item availablity

restricted.

Any correspondence concerning this service should be sent to the repository

administrator:

staff-oatao@listes-diff.inp-toulouse.fr

Suspension-Driven

gravity

surges

on

horizontal

surfaces:

Effect

of

the

initial

shape

N. Zgheib

a ,b ,∗

_,

_T.

_Bonometti

b

_,

_S.

_Balachandar

a

a Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611, USA b Institut de Mécanique des Fluides de Toulouse (IMFT) - Université de Toulouse, CNRS-INPT-UPS, Toulouse, France

Keywords:

Direct numerical simulations Geophysical flows Non-canonical Particle-laden Deposit

a

b

s

t

r

a

c

t

Wepresentresultsfromhighlyresolveddirectnumericalsimulationsofcanonical(axisymmetricand pla-nar)and non-canonical(rectangular)configurationsofhorizontalsuspension-driven gravitysurges. We showthatthedynamicsalongtheinitialminorandmajoraxisofarectangularreleaseareroughly sim-ilartothatofaplanarandaxisymmetriccurrent,respectively.However,contrarytoexpectation,we ob-serveundercertainconditionsthefinalextentofthedepositfromfinitereleasestosurpassthatfroman equivalentplanarcurrent.Thisisattributedtoaconvergingflowoftheparticle-ladenmixturetowardthe initialminoraxis,abehaviourthatwaspreviouslyreportedforscalar-drivencurrentsonuniformslopes [31].Thisflowisobservedtobecorrelatedwiththetravellingofaperturbationwavegeneratedatthe extremityofthelongestsidethatreachesthefrontoftheshortestsideinafinitetime.Asemi-empirical explicitexpression(basedonestablishedrelationsforplanarandaxisymmetriccurrents)isproposedto predictthe extent ofthedepositintheentire x-yplane.Finally,weobservethatfor thesameinitial volumeofasuspension-drivengravitysurge,areleaseoflargerinitialhorizontalaspect-ratioisableto retainparticlesinsuspensionforlongerperiodsoftime.

1. Introduction

Gravity currentsare primarily horizontalflowsthat are driven by a streamwise pressure gradient induced by the difference in densitybetweenthecurrentandtheambient.Ofparticular inter-est tothe presentwork are sediment-ladenor suspension-driven currents where the density difference is due to the suspended sediments. Animportant feature ofsuspension-driven currentsis their active interaction with the bed. Depending on the inten-sity anderosivepowerof thecurrentandon theavailability and size of sediments on the bed, a suspension-driven current can be in net-depositional, net-erosive or by-pass mode. In contrast to suspension-driven currents, scalar gravity currents, which are driven by temperatureor salinityinduced density difference, are conservative since the source of density difference is conserved overtime.

Bothscalarandsuspension-drivencurrentsareactivelystudied intheircanonical geometricconfigurationsofplanar and cylindri-calreleases(seeFig. 1 forschematicsoftheplanarandcylindrical releases).Thesecanonicalsetupshavebeenextensivelyresearched [14,16,18–22,24,27] resultingin the developmentof a widerange of

∗ _{Corresponding author: Department of Mechanical and Aerospace Engineering,}

University of Florida, Gainesville, FL 32611, USA.

E-mail address: nzgheib@ufl.edu (N. Zgheib).

simpleyet robustmodels. Indeed,one oftheadvantagesof deal-ingwithasimplegeometricconfigurationisthat attimesthe di-mensionalspacemaybe reducedtotwo orevena single dimen-sionmaking the derivationof elegant theoretical solutions possi-ble.One modelthat hasbeen particularlypopular isthe Navier– Stokes-basedshallowwaterequations[12,17,26,28] .

Along with the box model [16] , the shallow water equations form a powerful tool in the sense that they have the ability to producesimplealgebraicscalingrelationstopredictcertainkey as-pectsofgravitycurrents.Forexample,theextentofthedepositof a suspension-driven gravity surge initially confined within a cir-cular cylinder may be expressed in terms of the initial param-eters: cylinder dimensions, particle settling velocity, and particle volumefraction[4] .Asimilarrelationexists forthe extentofthe depositresultingfromaplanar,lock-releasesuspension-driven cur-rentaswell[5,10] .Thescalingrelationsthathaveprovenvery use-fulin the context of planar and cylindricalreleases are however notreadilyextendabletomorecomplexconfigurations.

Aplanaroraxisymmetricrelease(seeFig. 1 )isdefinedonlyby the initial vertical cross-section whose shape is characterized by the height to length (or height to radius)aspect ratio. For non-canonicalfinitereleases,thehorizontalcross-sectionoftherelease (ortheshapeoftherelease)mustbe additionallydefined.Incase of a rectangular or elliptichorizontal cross-section, these shapes canbecharacterizedintermsoftheinitiallengthtowidthaspect

Fig. 1. Schematic of a planar (left) and an axisymmetric (right) configuration. In the planar setup, the particle-laden mixture is initially confined behind a gate whereas in the axisymmetric setup, the mixture is initially confined inside a hollow circular cylinder.

ratio.Zgheibetal.[29] observedthatthehorizontalcross-sectional shapeoftheinitialreleaseisanimportantfactorwhichheavily in-fluences thespeed anddirectionof spreading.From both experi-mentsanddirectnumericalsimulations, theyobservedstrong az-imuthaldependencealongthecurrent-ambient interfaceinterms ofthefrontvelocity.Thisstudywaslaterextendedto suspension-drivengravitysurges[30] wheretheextentofthedepositwasalso observedto be significantlyaffected by theinitial shape. Intheir studythey assessed the importance of bedload transport in cor-rectlypredictingtheprofileofthedepositaswellastheshortand longtermeffectsoftheinheritinitialdisturbanceinlaboratory ex-periments.

The preferential spreading direction of non-planar and non-circularreleasesofarbitraryhorizontalcross-sectionsdepends pri-marilyontheinitial shapeoftherelease.However,inthecaseof gravitysurges spreading overan inclinedbottom surface[13,25] , thepreferentialspreadingdirectionisdictatedbythe presenceof theslope.There,the spreadingofa heavy fluid,initiallyconfined withinaslantedcircularcylinder,downauniformslopebreaks ax-isymmetry, and a three-dimensional self-similar shape was seen to evolve. Zgheib et al. [31] observed gravity surges on sloping boundariesto exhibit a convergingphaseof spreadingwherethe fluidneartheheadofthecurrentconvergedtowardthesymmetry plane.Duringthisphase,thevelocity withintheheadofthe cur-rentexhibitsastrongspanwisecomponent.Therelativeamplitude ofthisspanwisecomponentismagnifiedwithsteeper slopes,and vanishes,asexpected, whenthebottom inclinationbecomes hor-izontal.Thesefindings on theeffects of initialshape andbottom inclinationwillprovetobeimportantforthepresentanalysis.

Some of the real-world applications of non-canonical suspension-driven gravity currents include dredging, landslides, and building demolitions. In the case of dredging, it is often importantto knowhow the extent of the deposit relatesto the conditionsatthetimeofreleasesothattheaccurateplacementof thedredgedmaterialbecomespossible.Similarly,inthecaseofa controlled building demolition, the resulting debris cloud, which constitutesagravity surge, isinherently relatedto thegeometric propertiesofthebuildingamongotherparameters.

The purpose of the present study is to investigate the dy-namics of non-canonical suspension-driven gravity surges result-ing from rectangular initial releases of various horizontal aspect ratios. The currents to be considered are in the net-depositional regime and therefore after a well-defined period the suspended sedimentssettle onthe bedandthe currentdies.Based on scal-ingrelationsofplanarandcylindricalgeometries,wepropose sim-plesemi-empirical relations whichcan predict witha reasonable degree of accuracy the shape of the final extent of the deposit for non-canonical releases. We consider large scale simulations ofboth scalar-drivenandsuspension-drivengravitysurges,whose results are used to test the validity of the semi-empirical rela-tions.Intheremainderofthepaper,we willusetheterm

scalar-driven to refer to conservative/non-depositional currents (surges withzerosettlingvelocity), thetermsuspension-driventoreferto non-conservative/depositionalcurrents(surgeswithnon-zero set-tlingvelocity),andthetermgravitysurgetorefertoeither scalar-drivenorsuspension-drivencurrents.

Thefollowingsectionsareorganizedasfollows.InSection 2 ,we discusssomeofthesimplerelationsthathavebeenestablishedfor planarandaxisymmetricconfigurations.Themathematical formu-lation is briefly describedin Section 3 . In Section 4 , we present themainfindingsofthepapers.InSection 5 ,we proposea semi-empiricalexpressionforpredictingtheextentofthedepositfrom rectangular releases. This is followed by an analysis of different scenariosofreleasedependingonthehorizontalcross-sectional ra-tioinSection 6 .Finally,conclusionsaredrawninSection 7 . 2. Theoreticalestimatesforplanarandaxisymmetric configurations

Thetwo-layershallowwatermodel,whichisbasedonthe ver-ticallyintegrated Navier–Stokesequations both inthe heavy bot-tom current and in the ambient, was shown by Bonnecaze et al. [5] to offer a rigorous mathematical approach for the planar suspension-driven gravity surges. Dade and Huppert [11] devel-opedaboxmodelforhorizontalsuspension-drivencurrentsto ob-tainsimpletheoreticalestimatesofthehorizontalextentofthe de-posit. The characteristicthicknessof thedepositand therun-out time were compared against experimental measurements to re-markablesuccess(seealso[10] ).Therun-outtimemarksthetime whenthecurrentcomestoafull stopasaresultofall the parti-clessettlingout.Similarly,therun-outdistancecorrespondstothe distancetravelledbythecurrentbeforeitcomestoafullstop(i.e. themaximumextentofthedeposit).Additionally,the characteris-ticthicknessofthedepositrefers totheaveragedepositthickness overtheareawherethedepositoccurs.Thekeyresultsofthe pla-narboxmodelcanberecoveredwithascalingargument.

We consider afixed volumeV₀∗ ofrelease (perunit width) of particle-ladenfluidofinitialreducedgravityg₀∗definedas

g₀∗=g∗

φ

0

ρ

∗

p−

ρ

∗a

ρ

_a∗ , (1)

where

ρ

∗_pand

ρ

_a∗denotetheparticleandambientfluiddensities, respectively,and

φ

0istheinitialvolumefractionofparticlesinthe

current.Intheabove,andfortheremainderofthemanuscript,the asterisksuperscriptdenotesadimensionalquantity,andzero sub-scriptrefers to initialtime. Fortheplanar current,the volumeof release per unit width of thecurrent can be expressed in terms of the lock height and locklength as V₀∗=H₀∗ X₀∗. We now de-finethecharacteristicheightofthecurrenttobe H∗ andthe cor-respondinglengthofthecurrenttobe X∗=V₀∗/H∗,whichignores theeffectsofentrainmentordetrainmentofambientfluidor par-ticlesinto oroutofthe current.The frontvelocity ofthecurrent scalesas



g₀∗H∗andthecharacteristictimescaleofthecollapseis givenby thecharacteristic sedimentationtime T∗=H∗/

v

∗s,where

thesettlingvelocity v ∗

s isthat ofasingleisolated particlesettling

inaquiescentambientfluid.Thistimeroughlycorrespondstothe time required by a particle to crossa distanceequivalent to the initial/characteristic height oftherelease.We now usethe above scalingtoproperlynon-dimensionalizetheshallowwater concen-trationequation

∂φ

∂

t∗+u∗

_∂

∂φ

_x∗=−

v

∗s_h

φ

_∗. (2)

where u∗ isthe local horizontalvelocity in thecurrent,

φ

is the localvolumefractionofparticles andh∗isthelocalheightofthe current. In particular, we demand the resulting non-dimensional

equationtobeindependentof

v

∗

s,whichyieldsthecondition

v

∗ s H∗ =



g₀∗H∗ X∗ . (3)

This is equivalent to assuming that the characteristic time of sedimentation T∗ isof thesame orderofmagnitudeas the char-acteristictime ofpropagation X∗/



g₀∗H∗.SubstitutingforX∗ and rearrangingweobtainH∗=

(

v

∗2

s V0∗2/g _∗

0

)

1/5fromwhichwecan

ob-tain X∗ andT∗. The above scales can be shown to properly non-dimensionalize the shallow watermass and streamwise momen-tumbalancesaswell.Thus,inthecaseofaplanarcurrentthe ap-propriatescalingoftherun-outdistance(X_pd∗),run-outtime(T_pd∗), andcharacteristicdepositheight(H∗_pd)are

X_pd∗ ∼



g₀∗V₀∗3

v

∗2_s



1/5 , T_pd∗ ∼



V₀∗2 g∗ 0

v

∗3s



1/5 , H∗_pd∼

φ

0

φ

b



v

∗2 sV0∗2 g∗ 0



1/5 , (4) where

φ

0istheinitialvolumetricparticleconcentrationofthe

re-leaseand

φ

_b isthe final volumetricparticleconcentration within thedeposit,whichcan be takento bearound 0.5followingDade andHuppert[10] .Intheabovethesubscript“pd” standsforplanar deposit.Basedoncomparisonagainstexperimentalmeasurements andshallowwatersolutions,DadeandHuppert[10] recommended the constants ofproportionality in the above relations to be3,2 and1/3,respectively.

The scaling analysis for the axisymmetric suspension-driven surge waspresentedby Bonnecazeet al.[4] . Herethe volume of the cylindrical releaseis given by V₀∗=

π

R∗2₀ H₀∗,where againR∗₀

andH₀∗ aretheinitialradius andheight ofthecylindricalrelease. In anaxisymmetric currenttheradius ofthecurrent, intermsof thecharacteristicheightofthecurrent,goesas R∗=



V₀∗/

(

π

H∗

)

. The scalingof thefront velocityandtime remain thesameasin theplanarrelease.Weagainusethesamescales toproperly non-dimensionalizetheshallowwatergoverningequations,whichnow yieldsthecondition

v

∗ s H∗ =



g₀∗H∗ R∗ . (5)

Substituting for R∗ and rearranging we now obtain H∗=

(

v

∗2

s V0∗/

(

π

g _∗

0

)

)

1/4 from which we can obtain R∗ andT∗. The

ap-propriatescalinginthecaseofacylindricaldepositoftherun-out distance(R∗_cd),run-outtime(T_cd∗),andcharacteristicdepositheight (H_cd∗)are R∗_cd∼



g₀∗V₀∗3

v

∗2_s



1/8 , T_cd∗ ∼



V₀∗ g∗ 0

v

∗2s



1/4 , H_cd∗ ∼

φ

0

φ

b



v

∗2 s V0∗ g∗ 0



1/4 , (6) In the above the subscript “cd” stands for cylindrical deposit. Based on comparison against experimental measurements and shallow water solutions, Bonnecaze etal. [4] found the constant ofproportionalityfortherun-outradiustobe1.9.Inthecaseofa fulldepthreleasetheyalsoobservedtheabovescalingtobevalid onlywhentheinitialreleasesatisfiedthefollowingcondition

R∗₀



v

∗2 s g₀∗V*3 0



1/8 0.1. (7)

Instead,iftheinitialreleaseisinadeepambientthentheright handside ofthe aboveconditionchanges to0.3. Ifthe initial re-leaseviolatescondition(7 )thenthe run-outradial distancegiven in(6 )isonlyanupperboundandanempiricallowerboundis de-finedas[4] R∗_cd0.92



g₀∗3V₀∗7

v

∗6 s



1/18 . (8)

Fig. 2. Computational domain and initial shape of the release (here in the case χ0 = wl = 12 ). We consider a lock-exchange configuration, that is the initial height

of the gravity surge H ∗

0 is equal to that of the ambient fluid L ∗z , i.e. L z = 1 .

3. Mathematicalformulation

Weperformeda numberofsimulationsforwhicha sediment-laden mixture confined within a rounded-rectangular cross-sectional cylinder is released on a horizontal boundary (Fig. 2 ). Thesediment-ladenmixtureistreatedasacontinuumanda two-fluidformulationisadopted.WefollowCanteroetal.[7] by imple-mentingan Eulerian-Eulerian modelofthe two-phase flow equa-tions. We solve the conservation of mass and momentum equa-tionsforthecontinuumfluid phase,analgebraicequationforthe particle phase momentum, as well as the transport equation for the normalized sediment phase concentration field

ρ

. The non-dimensionalsystemofequationsreads

∇

· u=0, (9) Du Dt =

ρ

e g₋

_∇

_p+ 1 Re

∇

2_{u, (10)} up=u+

v

seg, (11)

∂ρ

∂

t +

∇

·

(

ρ

up

)

= 1 ScRe

∇

2

_ρ

_{. (12)}

Here up and u are the velocities of the particle and

contin-uumfluidphases, respectively.ImplicitinEq. (11) ,isthe assump-tionthatthe timescaleofthe particlesismuchsmallerthan the timeflowtimescalesandthustheparticlefaithfullyfollowsthe lo-cal fluid velocity exceptfor thevertical drift dueto gravitational settling[1,23] .Thesettling velocityvscorresponds tothe balance

betweentheStokes dragforce,actingon asinglespherical parti-cleassuming asmallparticleReynoldsnumber,andthebuoyancy force.Forthepresentsetofsimulations,vsisaninputanditsvalue

isprescribedinTable 1 .eg_{isaunitvectorpointinginthedirection}

ofgravity,andprepresentsthetotalpressurefield.ThevariablesSc

andRearetheSchmidtandReynoldsnumbersdefinedas

Sc=

ν

∗

κ

∗,Re=

L∗U∗

ν

∗ , (13)

where

ν

∗ represents the kinematic viscosity of the continuous phaseand

κ

∗istheeffectivemassdiffusivityoftheparticle-laden mixturein theambient fluid.The particle phase concentration is normalizedbetween0and1asshownin(14 ),andthelength, ve-locity,andtimescalesaredefinedrespectivelyas

⎧

⎪

⎨

⎪

⎩

L∗=H₀∗, U∗=

g∗

_φ

₀

ρ

∗p−

ρ

a∗

ρ

∗ a L∗_{, T}∗₌ L∗ U∗

ρ

=

ρ

∗−

ρ

a∗

ρ

∗ m0−

ρ

a∗ . (14)

Table 1

Details of the numerical simulations: The shape variable refers to the horizontal cross-sectional shape of the cylinder, which can be either planar (P), circular (C), or rectangular (R). χ0 is the initial cross-

sectional aspect ratio defined as the ratio of the total length ( l ) to total width ( w ) of the release. L x × Ly × L z and N x × N y × N z refer to the domain size and corresponding grid resolution, respectively. v s , Re ,

and Sc are the non-dimensional settling velocity, Reynolds number and Schmidt number, respectively. Sim # Shape χ0 l w Lx × L y × L z Nx × N y × N z vs Re Sc S1 R 12 5.68 0.47 18 × 13 × 1 960 × 694 × 159 0.02 8430 1 S2 R 8 3.78 0.47 18 × 12 × 1 960 × 640 × 159 0.02 8430 1 S3 R 12 5.68 0.47 20 × 16 × 1 1170 × 936 × 179 0 8950 1 S4 P ∞ ∞ 0.47 20 × 1.5 × 1 1086 × 82 × 159 0.02 8430 1 S5 C 1 0.47 0.47 7 × 7 × 1 380 × 380 × 159 0.02 8430 1 S6 P ∞ ∞ 0.47 20 × 1.5 × 1 1228 × 122 × 179 0 8950 1 S7 ∗ _{R 3.8 1.78 0.47 15 × 10 × 1 800 × 534 × 159 0.02 8430 1}

∗ _{Details of the simulation previously reported in Zgheib et al. [30]}

Intheabove,

ρ

∗and

ρ

_m∗₀representthelocalandinitialmixture densities,respectively. Here,we considerthe Boussinesq approxi-mationofsmalldensitydifferencebetweentheparticle-laden mix-tureandtheambientfluid inthattheconcentrationonlyappears inthe buoyancy termof the momentum equation (first termon therighthandside of(10 )).Eqs. (9 )–(12 )are solvedusinga spec-tralcode[8,9] withina rectangularcomputationaldomain shown inFig. 2 . Periodic boundary conditions are used for all variables in the streamwise, x, and spanwise, y, directions. At the bottom wall (z=0) no-slip and no-penetration boundary conditions and atthetopwall (z₌1)free-slipandno-penetrationboundary con-ditionsare imposed forthecontinuousphase velocity.Asforthe dispersedphase concentration field, Neumannandmixed bound-aryconditionsare enforcedatthe bottom(z=0) andtop (z=1) walls,whichtranslateintozeronetparticleresuspensionfluxand zeroparticlenetflux,respectively.



∂ρ

∂

z



z=0 =0;



1 ScRe

∂ρ

∂

z −

v

s

ρ



z=1 =0. (15)

Thelengthsofthedomaininthestreamwiseandspanwise di-rectionsarechosentoascertainthatthereisuninterrupted devel-opment of the gravity current. Details of the numerical simula-tionsaredescribedinTable 1 .ItwasdemonstratedinZgheibetal. [30] thatthepropagationofthecurrentisnotaffectedbythe lat-eralboundariesofthecomputational domainaslong asthefront ofthepropagatingcurrentisoneormorecurrentheightsfromthe boundary.Inallthesimulationcasesconsideredthehorizontal ex-tentofthecomputationaldomainwaschosento belargeenough tosatisfy the abovecriterion. The gridresolution for suspension-drivenandscalar-drivensurgesisthesameasthatusedinZgheib etal.[29,30] forrectangularreleases, wherethe adequacyofthe grid for converged solution has been established. Also the grid resolution employed is consistent with the requirement that the grid spacingmust be of the order ofO

(

ReSc

)

−1/2 _{[2,15] . We use}

two values for the Reynolds number Re=8430 and Re=8950 forsuspension-drivenandscalar-drivengravitysurges,respectively. TheSchmidtnumberissettooneforallsimulations.Thisis com-monpracticeinthesetypesofflows[23] .Furthermore,Bonometti &Balachandar[6] andNeckeretal.[23] demonstratethatthe ef-fectofthe Schmidtnumberon theflow intherangeofRe num-bersconsidered hereisnot importantaslong asit isof order1. Wethereforedonotexpect ourresultstobesensitive tothe pre-cisevalueofSchmidtnumber.TheseReandScnumbervalueswere alsochosentoallowforameaningfulcomparisonwithpreviously publishedresults(e.g. [29 ,30 ]).The timestep ischosen suchthat the Courantnumber remains below0.5. We impose a small ran-dom disturbance to the initial concentration field to stimulate a fastertransition to turbulence. The amplitude of the disturbance amounts to 5% of the density difference between the sediment-laden mixture and the ambientfluid. The initial interface ofthe

Fig. 3. Flow visualization of S1 using semi-transparent isosurfaces of concentration ( ρ) at three time instances, t = 3 (blue), 9 (green), and 20 (red). We observe strong azimuthal dependence for which the current is noticeably the weakest along the initial longer axis. A length scale of 4 non-dimensional units is shown above the isosurfaces. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

concentration field betweenthe sediment-laden mixtureand the ambientfluid issmoothenedsuch that thejump from0to1 oc-cursoverasmalldistanceof3gridcells.

4. Results

4.1. Rectangularvsplanaroraxisymmetricsuspension-drivengravity surges

Ina planar configuration,thegravity surgedoesnot haveany meanspanwisemotionandisrestrictedto flowinasingle direc-tion alongthe channel length.Similarly, inan axisymmetric con-figuration, the flow is radially outward withno meanazimuthal motion. In both cases the final deposit pattern remains statisti-callyplanarandaxisymmetric,respectively.Thethree-dimensional spatio-temporalevolutionofacurrentofinitialhorizontal rectan-gularshapeispresentedinFig. 3 .Thesurgeisvisualizedby mul-tiple semi-transparent isosurfaces of the concentration field

ρ

at threetime instances t=3, 9and20.The currentexhibitsstrong azimuthaldependenceandisthemost(resp.least)energeticalong theinitialshorter(resp.longer)axis.Inotherwords,for rectangu-larconfigurations,theinitialshapedictatesthepreferential spread-ing directionofthe current, andasa resultthe final deposit ex-hibitsazimuthaldependencyinboththeextentandtheareal den-sity.Inparticular,thecurrentextendsthefarthestalongits initial minor/shortaxisandpropagatestheshortestdistancealongits ini-tialmajor/longaxis.

For a scalar-driven gravity surge, Zgheib et al. [29] observed that atearly timesarectangularreleaseadvances alongits initial minoraxisataspeedequivalenttothatofaneffectiveplanar cur-rent if one takes the initial lock length of the latter as half the widthof the rectangularrelease, andthe height equalto that of

Fig. 4. Final extent of the deposit for the depositional cases in Table 1 . For a meaningful comparison with S5, the centre of the cylindrical release was shifted by −ηunits in the y -direction to coincide with the geometric centre of the rounded portion of the rectangle (circular dot in frames a, b, and c). The empirical lower bound in ( 8 ) for R cd

is needed because condition ( 7 ) is not met. (d) Maximum extent of the deposit (see frame b) along the x and y axes as a function of χ0 . For a meaningful comparison, 

has been shifted by −ηunits along the y -axis. For comparison, the maximum extent predicted by ( 4 ) for the planar χ0 = ∞ case is 6.0, while those predicted by ( 6 ) and

( 8 ) for the axisymmetric χ0 = 1 case are 2.6 and 1.7, respectively.

therectangularrelease.Ontheother hand,alongtheinitialmajor axis the current was observed to spread as an effective axisym-metriccurrentoflockradiusequaltohalfthewidthoftheinitial rectangularrelease,andofsimilarheight.

Inordertotestiftheaboveobservationsextendto suspension-driven gravity surges, fully resolved direct numerical simulations have been conducted here. These simulations are fully resolved onlyatthe macroscaleinthe computationofthe continuumEqs. (9) –(12 ),buttheydonotresolvetheflowatthemicroscalearound individual sediments.When the particlesettling velocity ismuch smallerthanthe characteristicvelocity ofthelargescaleflow, i.e.

v

∗

s



g₀∗H₀∗_, or equivalently vs  1, a suspension-driven

grav-ity surge is likely to behave as a scalar-driven surge to lead-ing order. The simulations consisted of suspension-driven rect-angular releases with different initial cross-sectional aspect ra-tios

χ

0, defined as the ratio of the longest to the shortest side,

alongwithsimulationsinplanarandaxisymmetricconfigurations. Somescalar-drivenreleaseswerealsoperformedforcomparison.It shouldbenotedthattheinitialcross-sectionalshapeoftherelease inthepresentstudyisarectangleinwhichtheshortedges have beenreplacedwithsemi-circles,thustheyareroundedrectangles. Thespreadingishowevernearlyidenticaltoatruerectanglewith right-angledcorners(see Fig.19in[29] ),andthereforethe subse-quent discussionis equallyapplicable to“true” right-angled rect-angles.

Fig. 4 showstheextentofthedepositforthesuspensiondriven rectangular surges andhow they compareto the planar and ax-isymmetricconfigurations.Theextentofthedepositinthe rectan-gular releasealong theinitial minor axis(x-axis) can be directly comparedwiththecorrespondingdepositforaplanarrelease.For ameaningfulcomparisonalongtheinitialmajoraxis,thecentreof theaxisymmetriccurrenthasbeentranslatedtocoincidewiththe centreofthecircularportionoftherounded-rectangular configura-tion.SeveralobservationscanbemadefromFig. 4: i)theextentof thedepositalongtheinitialmajoraxisagreesreasonablywellwith thatofanaxisymmetricreleaseirrespectiveoftheinitialhorizontal cross-sectionalaspectratio

χ

0.ii)Theextentofthedepositalong

theinitial minoraxisdependson

χ

0,andincreases as

χ

0 is

in-creased.iii)Forrelativelylargevaluesofthecross-sectionalaspect ratio(

χ

0 ࣡8) theextentofthedepositseemstoreacha sortof

plateau.iv)Theextentofthedepositalongtheinitialminoraxisof therectangularsurgesoflarge

χ

0,namely

χ

0=8and12,exceeds

thatoftheplanarreleasebyabout9and13%,respectively.Fig. 4 d showsthemaximumextentofthedepositalong thexandyaxes asafunctionof

χ

0forthedepositionalcasesinTable 1 .Whenthe

releaseisaxisymmetric (

χ

0=1),themaximum extentofthe

de-positisthesamealongthexandyaxes asexpected, howeveras

χ

0increases,thedistancetravelledbythecurrentalongthex-axis

appearstoincreasewhereasthatalongthey-axisseemstoremain unchanged. Interestingly,aswe approach theplanar limit (

χ

0 →

Fig. 5. Temporal evolution of (a) the front position and (b) the front velocity for all suspension-driven gravity surges in Table 1 (the legend in (b) applies to (a)). For rectangular releases, r N and u N are shown at both the initial minor ( y = 0 ) and

major ( x = 0 ) axes of the rectangular cross-section. The solid ellipse on the time axis in frame (a) corresponds to the time t = 6 where the front evolution of the χ0 = 12 and χ0 = ∞ currents begin to diverge from one another.

for

χ

0=8and12by9and13%,respectively.Wewilladdressthis

somewhatunexpectedobservationinSection 6 .

We may now lookat thelocal dynamics ofthe front of rect-angularsurgesatsome specificlocations,namelyalongtheinitial minorandmajoraxes.Fig. 5 showsthetemporalevolutionofthe travelleddistancealongthemajorandminoraxesforsomeofthe simulationsshowninTable 1 .Thedynamicsofthefrontalongthe initialmajoraxisisobservedtobenearly independentof

χ

0 and

inverygoodagreementwiththatofanaxisymmetricreleaseboth interms offront position andvelocity, asalready shownfor the finalextent ofthedepositinFig. 4 .Notethat thisisinlinewith thedynamicsobservedforscalar-drivengravitysurges(seee.g.Fig. 14bin[29] ).

The dynamicsofthefrontalongthe initialminoraxisismore complicated. Consideringthe frontvelocity (Fig. 5 b), one can see thatthedynamics issimilarinthe accelerationphaseup totime

t=1 while in the slumping phase, the mean value of the front velocity uS andthe corresponding duration tS depend on

χ

0. In

particular,uSdecreasesfrom0.45to0.40whiletSincreasesfrom3

to5,approximately,when

χ

0 isincreasedfrom3.8to12.Beyond tS,allthegravitysurgesexperienceadecelerationphaseofroughly

similartrendbutforwhichit isdifficulttodrawanydefinite de-pendencyrelativeto theinitialhorizontalaspectratio.Recallthat thepresentfrontvelocityiscomputedataspecificlocationofthe frontcontour, andhence theinstantaneous variation of thefront position,dueto theevolution ofthe lobesand clefts,is likelyto introduce some fluctuations in the front velocity, asobserved in Fig. 5 b.Thesubtlebutnoticeableinfluenceof

χ

0observedforthe

frontvelocity is alsovisiblein the plotsof thetime evolution of the front position (Fig. 5 a). For instance, the front extent of the

χ

0=3.8 surge isslightly above the othersfor times t ࣠6,then

thatofthe

χ

0=8surgeisthelargestuptot࣠18andfinallythe

χ

0=12surgeovercomestheothersatlatertimes.

Thecomparisonoffront dynamicsalong theinitialminoraxis withthat of the planar surgeis somewhat atypical since the in-stantaneousfrontextentoftheplanarsurgeisclosertothatofthe

χ

0=3.8surgeratherthanthatofthe

χ

0=8and

χ

0=12surges.

Thisisinlinewiththeobservationmadeforthefinalextentofthe depositinFig. 4 d.Again,thispointwillbeaddressedinSection 6 .

4.2. Convergingflowinrectangulargravitysurges

As observed in Figs. 4 d and 5 a, it is somewhat surprising thatthefiniterectangularreleasesextendfartherorevenadvance fasterthanaplanar release.Indeed,one maywonder whatisthe mechanism by which a finite release, whose planform area in-creases quadratically with size, advances faster or even extends fartherthanaplanarreleasewhoseplanformincreaseslinearly.A possible explanation can be proposed from theresults presented in Fig. 6 , where we plot the x and z integrated concentration field,hx(y,t),definedas hx

(

y,t

)

=  Lx 0 Lz 0

ρ

(

x,y,z, t

)

dzdx, (16)

asafunctionofyforthesuspension-driven(S1andS4)and scalar-driven(S3andS6)gravitysurges.Fortheplanarrelease,wefurther averagehx inthehomogeneousy-direction anduseitfor

compar-isonwiththe rectangularreleases.Because ofintegrationalong x

andz,theprofilesinFig. 6 arelesssusceptibletoturbulent fluctu-ationscomparedtothelocalconcentrationfield(seeFig. 3 ). How-ever,inthecaseofparticle-drivensurges,asmoreandmore sed-iments deposit on the bottom wall and exit the computational domain, the mean value of the concentration decreases. Conse-quently, the fluctuationsinthe concentration field aboutthe de-cayingmean concentration value become more pronounced. This couldexplainthestrongsymmetry(abouty=0)atearlytimesin theprofilesofFig. 6 aandthemoderatedeviation,fromsymmetry, atlatertimes.Ontheotherhand,becauseoftheconservative na-tureoftheflowforscalar-drivencurrents,theprofiles inframeb remainverysymmetricuptothetimesconsideredherein.

First,we noticethat the meanvalue ofhx(y, t) decreases over

time asa resultofsedimentation.Itisalsoclearthat therelative amountofdensefluidgraduallyincreaseswithtime atthecentre plane (y=0) above that ofthe corresponding planar release.For instance,themaximumvalueofhx is2%,11%,and42%higherthan

thecorrespondingaverageoftheplanarsuspension-drivengravity surgeattimest=2,6and10,respectively. Thismayindicatethe presence of a converging flow toward the initial minor axisand supportstheobservationregardingthefasterspreadingandlonger depositionalextentoffinitereleases(seee.g.Fig. 5 ).Asubtlepoint that should be stressed is that the converging flow initiates at thetwoextremitiesoftherectangleandslowlymakesitsway to-wardtheinitialminoraxis(y=0).Onemayobservethatat t=2,

hx(y, t) in thevicinity of the initial minoraxis(|y| < 0.5) forS1

(

χ

0=12) isnearlyidenticaltothemeanvalue obtainedfromthe

planarcase(S4),whilehx(y,t)for

|

y

|

=1.5isnoticeablylargerthan

themeanplanarvalue.

Thesamebehaviourisobservedforscalar-drivengravitysurges (Fig. 6 b).Aconvergingflowwasalsoobservedinthecaseofsurges spreading on sloping boundaries [31] . In that case, a relatively strong cross-flow velocity component relative to the streamwise velocitycomponentwaspresentandtheconvergingflowwas eas-ilyidentifiedusingavectorplotofthevelocityfieldintheheadof thegravity current. Thisis not thecase, however,forrectangular initial releases of suspension-driven gravity surges on horizontal boundarieswhere themagnitude ofthe cross-flowvelocity com-ponent is about an order of magnitude smaller than that of the streamwisecomponent(notshown).Instead,inFig. 7 ,wepresent isocontoursofamodifieddepth-integratedycomponentofthe

ve-Fig. 6. x - z integrated concentration field h x vs y at t = 0 () , 2( ), 4( ), 6( ), 8( ), and 10( ) for (a) suspension-driven [S1 (solid lines) & S4 (dashed lines)] and (b) scalar-

driven [S3 (solid lines) & S6 (dashed lines)] gravity surges. The symbols mark the spanwise locations y where h x of the χ0 = 12 current is equal to that of the χ0 = ∞ planar

current.

Fig. 7. Contours from S1 in the x - y plane of the depth-integrated (within the current) y component of the instantaneous velocity field v ρ(see text for definition). Regions

in red corresponds to v ρ ≥ 5 × 10 −3 and indicate a converging flow toward the y = 0 -plane whereas regions in blue corresponds to v _{ρ≤ −5 × 10} −3 and indicate a diverging

flow away from the y = 0 plane. The outer black solid line in each frame corresponds to the location of the front and represents the isocontour h = 5 × 10 −2 where h is the

vertically integrated density field defined as h =  Lz

0 ρdz . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this

locityfieldv_ρinthex-yplane.Ineachofthe4framesinFig. 7 ,the isocontoursofv_ρ areboundedby anoutersolidblackline,which correspondstothelocationofthefront.Here,v_ρ isdefinedas

vρ

(

x,y,t

)

=−y

|

y

|

 Lz

0

ρ

vdz, (17)

wherevrepresentstheycomponentofthethree-dimensional ve-locityfield u.Therefore, inregions where v_ρ >0, theflow con-vergestowardthesymmetryplane y=0andviceversa.Atearly timest=2,we cansee thepresence oftwo zonesofconverging flowintheregion 1≤ |y| ≤ 2within thesuspension-driven grav-itysurge. As time evolves, theseregions spread andtend to ap-proachthesymmetryplane,suchthatattimet₌5,theflowinside the currentlocated inthe region close to the initial minor axis, i.e.|y| ≤ 2is mostlyorientedtoward thesymmetry plane y=0. This observationconfirms the presence of a converging flow to-wardtheinitialminoraxiswhichmayberesponsibleforthemass buildupobservedinFig. 6 initiatingattheextremitiesand propa-gatingtowardtheplane y₌0.Consequently,suchamassbuildup maymodify thelocal front velocitysince the latterscales as the square-rootofthefrontheight[3] .

4.3.Travellinginformationalperturbationalongthefrontcontour

Onewaytotrackhowinformationtravelsalong thefrontisto seehow certain parameters that are otherwise uniform in space (alongthe y-axis) fora planar release spatially evolve when the releaseisoffinitesize. Thenormal-to-the-frontangle(ϑ) andthe normal-to-the-front velocity (uN) are two such parameters. We

chooseϑ asone ofthetwo parameters becauseitprovides infor-mationon theinstantaneous localspatialandtemporalevolution ofthefront. Anotherparameteranalogousto _{ϑ that}could poten-tiallyprovideequivalentinformation isthelocal curvatureofthe front. However, to evaluate ϑ only first order derivatives are re-quired,unlikeevaluationofthelocalcurvature.Here,_ϑ(y,t)is com-putedas

ϑ

(

y,t

)

=arctan

(

∂

xN/

∂

y

)

, (18)

where,xNcorrespondstothefrontpositionintheuppersymmetry

plane(x≥ 0).Atthetimeofreleaset=0,ϑ(y,0)issuchthat

|

ϑ

(

y,0

)

|

=0 for

|

y

|

<

η

,

|

ϑ

(

y,0

)

|

>0 for

|

y

|

>

η

. (19)

where

η

isthedistancefromthe geometriccentre ofthe rectan-gular cross-section to the local centre ofthe rounded portion of therectangle(seeFig. 4 ).Tomonitorwhentheflatportionofthe frontceasestoremainflat,weneedtoseewhen

|

ϑ|

>



,where



isasmallnumber.Wethereforeneedasuitablevalueof



thatis small,toaccuratelydetectthechangeinthedirectionofthefront asearly aspossible. However,



must be large enough to obtain aclearsignal-to-noiseratio.Wefindforthe presentproblem,the isovalue

ϑ

=0.1to bea reasonablechoice.As forthe normal-to-the-frontvelocityuN thevalue uN=0.39ischosen asareference

sinceitisclosetothevalue ofthemeanslumpingvelocityofthe planarcurrent (see Fig. 5 b,S4). This islikely to bethe slumping velocitythattheflatportionoftherectangularreleaseexperiences beforetheinformationonthe“finiteness” ofthereleasereachesit. Fig. 8 aandbshowcontoursof|_ϑ|anduN,respectively,inthey-t

planeforearlytimest≤ 4,whereasFig. 8 coverlaystheisocontours of

ϑ

=0.1anduN=0.39ontopofthefrontpositionintheupper

symmetryplane (x_{≥ 0).} Astime evolves, theregionalong which thefront remainsflat(|ϑ|≤ 0.1)andadvancesataspeedcloseto thatoftheplanarrelease(uN≥ 0.39)decreases.Therateatwhich

theflatportionofthereleasedecreasesisroughlyconstantatearly times(t≤ 4).FromFig. 8 a,wecanestimatetheslope dy/dtofthe

|

ϑ|

=0.1 isocontourto be dy/dt ≈ 0.23.On the other hand, the

Fig. 8. Contours in the y - t plane from S1 of (a) the absolute value of the normal-to- the-front angle ϑ = tan −1_{(∂ x N /∂y) and (b) the normal-to-the- front velocity u}

N( y , t ). (c) Temporal evolution of the front in the upper symmetry plane ( x ≥ 0) at equal increments in time of ( t = 0 . 1 ). The thick blue and red lines correspond to the isocontours of |ϑ| = 0 . 1 and u N = 0 . 39 , respectively. Inset shows schematic of ϑ on

the curved portion of the front. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

slope dy/dtoftheuN=0.39isocontourisestimatedfromFig. 8 b

tobe dy/dt≈ 0.32.

The isocontours of

ϑ

=0.1 anduN=0.39are easily extracted

at early times (t _{≤ 4)} when the front is still relatively smooth. Atlatertimes,thefrontbecomesmorecomplexandhighly three-dimensional duetothe lobeandcleft instability,which makes it moredifficulttopreciselyextracttheaboveisocontours.However, we canobtaina timeestimate (tp,dy/dt)ofwhen theperturbation

reaches the symmetry plane (y=0), by assuming that the slope

dy/dt remains constant asthe perturbation advances inward. We shouldnote nonethelessthat thisassumptionmaynotnecessarily be true,as theperturbation speed may changeasit advances. A betterestimatemaybeobtainedbymarkingthetime(tp,rN)when

the front position rN(t) along the x-axis from S1, S2, andS7

ex-ceedsthatfromS4.FromFig. 5 a,wefindtp,r_N≈ 2,3,and6forS1,

S2,andS7,respectively.Acomparisonofthecharacteristic pertur-bationtimesaswellasthecharacteristicsedimentationand propa-gationtimesareshowninTable 2 .Theobservedperturbationtime

tp,r_N issmallerthantheestimatedtimetp,dy/dt,whichsuggeststhat

the perturbation speed does not remain constant, but rather in-creasesatlatertimes(t>4)asitadvancesinward.

In all the simulations considered herein, the perturbation reachesthe symmetry plane (y₌0) before the currentstops. In-deed,astheperturbationandpropagationcharacteristictimesare ofthesameorder,weshouldexpecttheconvergingflowtoreach the symmetryplane before thecurrentarrives ata standstill. On theotherhand,ifthehorizontalaspectratioofthereleaseisvery large,i.e.

χ

0 1,thecharacteristicperturbationtime willbe

cor-respondinglyverylarge aswell, whereasthecharacteristic propa-gationtime willapproachthatoftheplanarrelease(S4).Forsuch a case, the converging flow will have no effecton the dynamics ofthecurrentalongthe symmetryplane (y=0). Asforthe char-acteristicsedimentationtime, we observeTsto be aboutan order

ofmagnitudelargerthantheperturbationtimestp,dy/dtand tp,r_N.

Thisimpliesthattheeffectofsedimentationisnotlikelytobean importantfactorinthedynamicsoftheconvergingflow beforeit reachesthesymmetryplane.

Overall, Fig. 8 clearlyshowsthat the informationofthe finite size ofthe releasealong they-direction propagates fromthe cor-ners to the centreline (y=0) along the front. The precise value ofthevelocityatwhichthisinformationpropagates,however,and

Table 2

Characteristic perturbation, sedimentation, and propagation times from S1, S2, S4, and S7.

Sim # χ0 Characteristic perturbation time # Characteristic perturbation time £ Characteristic sedimentation time Characteristic propagation time tp,dy/dt = [ l ∗/ ( d y ∗/d t ∗) ] / T ∗ tp,rN Ts = ( H 0∗ / v∗s) / T ∗ Tp = [ max (∗, x ) /  g∗ 0 H 0∗ ] / T ∗ S7 3.8 [ 2 . 8 − 3 . 9 ] 2 50 5.2 S2 8 [ 5 . 9 − 8 . 2 ] 3 50 6.5 S1 12 [ 8 . 9 − 12 . 3 ] 6 50 6.9 S4 ∞ ∞ – 50 6.0

# Both values of dy / dt are considered: d y/d t _|

ϑ=0.1 and d y/d t |uN=0.39 £ _t

p,rN is the time when the front position along the x -axis from S1, S2, and S7 exceeds that from S4

Fig. 9. Retention rate of particles in suspension as a function of time. For the same initial volume of a suspension-driven gravity surge, larger cross-sectional aspect ra- tio releases are able to retain particles in suspension for longer periods of time.

its dependenceon other parameters remain unclear.It should be noted that the slope dy/dt of theuN=0.39 isocontouralso

pro-vides some measure for how informationtravels within the cur-rent.The frontvelocity uN isrelatedtothe current’sthicknessor

height,hN,bytheFroudecondition

uN ∼



hN, (20)

andthereforeasuN variesspatially,somusthN.However, uN isa

mucheasierquantitytoevaluatethan hN,becauseunlike uN,there

isnocleardefinitionforthefrontheight.

4.4. Particlerateofretentionasafunctionof

χ

0

Oneaspectthatisofgreatinterestinthestudyof suspension-drivengravitysurgesisthedepositionalrate.Itisoftenveryuseful to know how long a suspension-driven current can maintain its particles insuspension beforecomingto afull stop.In Fig. 9 ,we plot thenormalized amountofparticles remaining in suspension overtime,

β

,forall suspension-drivensimulationsinTable 1 .

β

is definedas

β

(

t

)

= ∫ Lx 0 ∫ Ly 0 ∫ Lz 0

ρ

(

x,y,z,t

)

dzdydx ∫Lx 0 ∫ Ly 0 ∫ Lz 0

ρ

(

x,y,z,0

)

dzdydx . (21)

Asexpected, circularreleaseshavethelowestretentionrateof particles,whereasplanarreleaseshavethelargest.Thisisadirect resultofthe everdivergingspreading natureofaxisymmetric re-leases ascomparedto theunidirectional spreadingofplanar cur-rents. Inaddition, forthe sameinitial volume ofa particle-laden mixture, releases with large cross-sectional aspect ratios,

χ

0,

re-taintheirparticlesinsuspensionforlongertimes.Forinstance,the value

β

₌0_.15,whichcorrespondstothefactthat15%ofthe par-ticlesremaininsuspensioninthegravitysurge,isreachedattime

t=12.1and15.6for

χ

0=1and12,respectively.Similarly,attime t=10,approximately24%(resp.34%)oftheparticles remain sus-pendedinthegravitysurgeofhorizontalaspectratio

χ

0=1(resp.

12 ).

5. Predictivemodelsforrectangularsuspension-drivengravity surges

5.1. Finalextentofthedeposit

AsshowninFig. 4 a, theshallowwatertheoreticalexpressions (4 )–(6 ) and the empirical relation (8 ) predicting the final extent of the deposit for the planar and axisymmetric releases, agree wellwiththepresentplanarandaxisymmetricsimulations.For in-stance,intheplanar case,the extentofthedepositfrom(4 )and fromthespanwiseaverageofS4are5.7and6.0,respectively. Sim-ilarly, forthe axisymmetric release,the azimuthal average of the extentofthedepositS5 is2.0,whichliesbetweenthelowerand upperboundsof1.7and2.6fromexpressions (8 )and(6 ), respec-tively.Here, theempirical lower bound is used becausethe con-ditionin (7 ) isnot met. In addition,these expressions provide a reasonablygood estimate forthe extentof thedeposit along the initialminorandmajoraxesoftherectangularrelease.

Here we propose an estimate of the extent of the deposit in theentire x-yplane. Zgheib et al.[29] showedthat scalar-driven rectangularreleasesreachaself-similarspreading phaseinwhich theazimuthalvariationofthefrontpositionmaybedescribedby asinusoidal functionof thelocal angle

θ

inthe polarcoordinate system,whoseorigincoincideswiththegeometricalcentreofthe release,

θ

beingmeasuredanticlockwisefromthex-axis.We there-foresuggestasimilarsinusoidalfunctiontodescribetheextentof thedepositwiththehelpof(4 ),(6 )and(8 ).Theempirical expres-siondescribingthe extentofthe finaldeposit



whichis the ra-dial distance betweenthe local front and the geometrical centre reads

(

θ

)

=Asinn

(

θ

)

+B, (22) with A=

X ∗ pd−



R∗_cd+

η

∗



H₀∗ and B=



R∗_cd+

η

∗



H₀∗ . (23)

where

η

∗isthedistancefromthegeometriccentreofthe rectan-gular cross-section to the local centre of therounded portion of therectangle (see Fig. 4 ), X∗_pd isgiven by (4 ), andsince the con-ditionin (7 ) is not met, the upperand lower bounds of R∗_cd are givenby (6 ) and(8 ), respectively. Here again, the asterisk super-scriptdenotesadimensionalvariable,allothervariablesaretobe understoodasnon-dimensional.Weusethescalinggivenin(14 )to non-dimensionalize(22 ).Theexponentnin(22 )istheonlyfitting parameterofthefunction.

Fig. 10 ashowstheextentofthefinaldeposit



asafunctionof

θ

forS2alongwiththetheoreticalandsemi-empiricalpredictions ofR∗_cd andX_cd∗.Thevalue n₌6wasobservedtoprovidethebest agreement.Othervaluesofnintherange[4,8]gaveroughly simi-larresults.Similarly,Fig. 10 canddshowtheextentofthedeposit inthex-y plane forS1,S2 andS7 aswell as



asobtainedfrom (22 ).ItshouldbenotedthatforthecasesconsideredinFig. 10 ,the conditionin(7 ) doesnot hold,andR∗_cd isboundedby the

condi-Fig. 10. a) Extent of the final deposit vs the polar coordinate θfrom S2 (thick black line). The pink shaded region is the prediction from ( 22 ) using both bounds of the axisymmetric release from ( 6 ) and ( 8 ). For comparison, the prediction from ( 4 ) of the final extent for a planar gravity surge (green horizontal dash-dot line) and those from ( 6 ) and ( 8 ) for an axisymmetric gravity surge (blue horizontal long dashed lines) are also plotted. (b)-(d): Extent of the final deposit in the x - y plane from S7 ( χ0 = 3 . 8 ), S2 ( χ0 = 8 ), and S1 ( χ0 = 12 ) respectively. The dashed red lines

correspond to the prediction from ( 22 ) using the upper and lower bounds from ( 6 ) and ( 8 ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

tionsin(6 )and(8 )[4] .Goodagreementisobservedinall3cases, therelativediscrepancybetweenthe simulatedandpredicted ex-tentsofthedepositalongthexaxisbeingapproximately_−17, 9_, and 13%, for

χ

0=3.8, 8 and 12, respectively. Along the y axis,

theextentofthedepositfromthesimulations liewithinthe pre-dictedboundsandare closertothe empiricallower bound given in(8 ).

5.2.Horizontalaspectratioofthedeposit

Using(22 ),onemayalsopredictthefinalaspectratioofthe de-posit

χ

∞.Notingthatwiththepresentcoordinatesystemthe mini-mumandmaximumextentofthedepositcorrespondto

θ

=0and

θ

=

π

/2, respectively, we have

χ

_∞=

(

A+B

)

/A=X_pd∗/

(

R∗_cd+

η

∗

₎

_.

Usingthefact that

χ

0=

(

R∗0+

η

∗

)

/R∗0,we caneliminate

η

∗ inthe

expressionfor

χ

_∞ andobtain apredictionof thefinal horizontal aspect ratioof the deposit as a function of the initial geometric parameters,viz

χ

∞=

X_pd∗ R∗_cd+R∗₀

(

χ

0− 1

)

.

(24) Aplotof

χ

∞ vs

χ

0 forthepresentsetofparametersisshown

inFig. 11 .Forsmallvaluesof

χ

0,(24 )performspoorlyasexpected.

Indeed, as the initial cross-sectional aspect ratio tends to unity, one would expect both the “minor” and“major” axis of the re-leaseto behaveidenticallyasanaxisymmetric currentandhence

χ

∞ ≈ 1.Incontrast,(24 )gives

χ

∞≈ Xpd∗/R∗cd whichislargerthan

unitysince withthe presentmodel, theinitial minor axisof the rectangularreleasecan only exhibit a planar-like type of spread-ing.Ontheotherhand,forthepresentsetofparameters,as

χ

0 is

increased,(24 ) agreesverywell withexperimentalandnumerical data.Letusrecallthat(24 )isapplicablewhenever(4 ),(6 )and(8 ) arevalid,i.e.toamuchwidersetofparametersthaninthepresent case.

Fig. 11. Final deposit aspect ratio χ∞ vs χ0 for all the suspension-driven simula-

tions (black square symbols) in Table 1 . The blue circle corresponds to experiment 1 in Zgheib et al. [30] . The experiment is equivalent to S7 in Table 1 . The shaded region corresponds to ( 24 ) using the upper and lower bounds in ( 6 ) and ( 8 ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

6. Scenariosofdynamicsanddepositionofa suspension-drivengravitysurge

Here,weproposeascenarioforthedynamicsofa suspension-driven gravity surge of initial arbitrary shape depending on the competition between the characteristic times of sedimentation, propagation,andreflectionoftheperturbation.

We may now consider the planar and axisymmetric configu-rations as limiting cases for rectangular cross-sectional releases. When the initial cross-sectional aspect ratio

χ

0=1, the

rectan-gularreleasereducestoanapproximateaxisymmetriccircular re-lease. Onthe other hand,as

χ

0 → ∞, weretrieve theplanar

re-lease. All other rectangular releases with finite valuesof

χ

0

ex-hibitabehaviour thatisacombinationofbothcanonicalreleases. Thatisthebroadedgeoftherectangularcross-sectionbehaves,at leastforsomefiniteinitialtime, asa planarcurrent, whereasthe shortedgeactsasa divergingaxisymmetricreleaseuntilthe cur-rentcomestoafullstop.

Thefrontalongtheinitialminoraxisadvancesasaplanarfront untiltheinformationofthe“finiteness” ofthereleaseisreceived. ThiscanbeseeninFig. 5 aastowhenthefrontlocationsfor

χ

0=

8 and 12 separate fromthat ofa planar current. Thisstatement implies that there is an information wave that travels along the front of the currentand carries withit the information that the releaseis of finitelength in they-direction anddoesnot extend indefinitelyalongtheinitialmajoraxis.

The reason the final extent of the deposit along the minor and major axes of the rectangular release conforms well to pla-nar and axisymmetric releases is in fact due to the fronts along thosespecific directionsbehavinginasimilar fashiontotheir re-spective canonical currentsfrom thetime ofthe releaseonward. Fig. 5 showsthetemporalevolutionofthetravelleddistancealong the major andminoraxes forsome ofthe simulations shownin Table 1 .Thefollowingobservationscanbemade.

i) Itconfirmsthatthetemporal evolutionalongtheinitial major axisisequivalenttothat froman axisymmetricreleaseandis nearly independent of

χ

0. The samebehaviour wasobserved

forscalar-drivencurrents(Figure14bin[29] ).

ii) Thereappears tobe some criticalinitialcross-sectional aspect ratio

χ

0cr1 abovewhichtheextentofthedepositexceedsthat

froman equivalentplanar current. Forthe present set of pa-rameters,weobserveavalueof

χ

0cr1≈ 5.

iii) There exists another higher initial critical aspect ratio

χ

0cr2,

above whichthe extentofthe depositalong the initialminor axisbecomescompletelyindependent of

χ

0 andmatches

ex-actlytheextentfromanequivalentplanarrelease.

χ

0cr2

corre-spondstoaninitialreleaseforwhichtheconvergingflownever reachesthesymmetryplane(y=0).

iv) There exists an initial aspect ratio

χ

0max such that

χ

0cr1 <

χ

0max <

χ

0cr2 forwhichthe depositattains its largest

possi-bleextentalongtheinitialminoraxis.

v) For values of

χ

0 below

χ

0max, the current (along the initial

minoraxis)experiencesalargerslumping(nearlyconstant) ve-locityforsmallervaluesof

χ

0,butthedurationofthisnearly

constantvelocityphaseincreasesas

χ

0 increases(seeFig. 5 b).

Thesamebehaviourwasalsoobservedforscalar-drivengravity surges(Figure14bin[29] ).

vi) For values of

χ

0 such that

χ

0cr1 <

χ

0 <

χ

0cr2, the planar

frontvelocityprovidesalowerlimittothefrontvelocityalong theinitialminoraxisfortheentiredurationoftheflow.Indeed asseenfromFig. 5 b,thefrontvelocity alongtheinitial minor axisfromS1andS2isalwayslargerthanorequaltothe corre-spondingplanarreleaseofS4.

Indeed, as

χ

0 attains larger and larger values, there comes a

point forwhich the currentcomes toa full stop before the con-vergingfloweverreachesthesymmetryplane(y=0).Inthatcase, the front along the initial minor axis advances as a planar cur-rent. Furthermore, asdiscussed in point (v), rectangular releases withvaluesof

χ

0 suchthat 1<

χ

0 <

χ

0max,experiencealarger

slumping(nearlyconstant)velocityforsmallervaluesof

χ

0.This

maybeduetothefactthatforsmallvaluesof

χ

0,theconverging

flow reachesthe initialminoraxisearlierwhen thecurrentis al-readyrelativelymoreenergeticthanatlatertimes.Theconverging flowwouldfurtherincreasethethicknessofthecurrentand there-foreraisetheslumpingvelocityalongtheinitialminoraxis(recall thatuN∼ h1N/2).

7. Conclusions

We presented results from highly resolved direct numeri-cal simulations of finite release horizontal gravity surges, both suspension-drivenandscalar-driven.Canonical(axisymmetricand planar)andnon-canonical(rectangular)configurations were stud-ied,whichallowedustovarytheinitialhorizontalcross-sectional aspect ratio

χ

0 overa widerange.The purposeofthestudywas

tobuild ontheknowledgeregardingaxisymmetricandplanar re-leases and useit to propose simple relations for certain key pa-rametersofrectangularconfigurations.

We showedthat the final extentof thedepositalong the ini-tialminor(resp.major)axisofaninitiallyrectangular suspension-driven gravitysurgeresemblesthat ofa planar(resp. axisymmet-ric)releasewiththesameinitial heightandhalfthewidthofthe rectangularrelease.Inaddition,rectangulargravitysurgeswiththe same initial volume but withlarger

χ

0, retain their particles in

suspensionforlongertimescomparedtosurgesofsmaller

χ

0.

Apredictionforthefinal extentofthedepositintheentirex

-y plane (and not justalong the major andminoraxes) was pro-posedusingavailabletheoretical andempiricalmodelsfor canon-icalsuspension-drivengravitysurges.FollowingZgheib etal.[29] , we proposeda sinusoidalempiricalrelation(22 ),forthefinal ex-tent of the depositasa function of azimuthal orientation whose amplitudeisobtainedfrompreviouslyestablishedrelationson ax-isymmetric andplanarcurrents.Additionally,asimpleexplicit ex-pression (24 ), relatingthe finalaspect ratioofthedeposit,

χ

_∞ to the initial cross-sectionalaspect ratio ofthe release,

χ

0 was

pro-posed. While the present simulations are based on specific val-uesforthe settling velocityaswell asthe ReynoldsandSchmidt numbers, theproposed semi-empirical expressions (22 ) and(24 ), because oftheir inherentcoupling withEqs. (4) ,(6 ), and(8 ),are applicable to a wide range of parameters includingthe Reynolds andSchmidt numbers,theheight andcross-sectionalaspectratio of the release, the particle settling velocity, the particle volume fraction, aswell as theinitial reduced gravity. Therefore, the

ex-pected effects of changing these parameters may be directly in-ferred from thecanonical, semi empirical expressions of (4 ), (6 ), and(8 ).Therefore,withtheknowledgeoftheinitialconditionsof therelease,itispossibletopredicttheextentofthefinaldeposit ofa rectangularrelease.Conversely, using(22 ) and(24 ), one can choosetheinitialparametersofthereleasetocontrolthefinal ex-tentofthedeposit.

Oneofthe mostcounterintuitiveoutcomesis thatcontrary to expectation, some suspension-drivengravity surges may advance slightly but noticeably faster and extend farther than equivalent planarsurgesthatexclusivelyundergounidirectionalspreading(up to 13% of the extent for the present range of parameters). In-tuitively, one would expect the final extent of the deposit from planar gravity surges to act as an asymptotic limit to the extent ofthe deposit along the initial minoraxis of rectangular gravity surges. This is shown not to be the case. This unexpected out-come is attributedto a convergingflow inside the surge toward theinitialminoraxisoftherectangularcross-section.Aconverging flow behaviour wasalso observed forfinite release scalar-driven gravity surges on uniformslopes [31] ,the intensity ofwhich in-creasessharplyforsteeper slopes.Thisleadsustoconjecturethat forsuspension-drivengravitysurges releasedon a uniformslope, theextentofthedepositalong theinitialminoraxisofa rectan-gular release might significantly exceed that of a planar release. Thediscrepancyisexpectedtobefurtherhighlightedastheslopes becomesteeper.

Thisconvergingflowisbroughtuponbyanon-uniformspatial distributiondueto thefinite natureof thereleaseasopposed to planarreleasesthatextendindefinitelyalongthey-axis.By track-ing the spatiotemporal evolution of otherwise spatially uniform variables for planar releases, we provided some estimate for the speedofpropagationoftheinformationalwaveandthe character-istictimeofreflectionatthesymmetryplane.Wedonotcurrently knowwhat parameters control oraffectthe speed ofthe pertur-bationwave.Thisisaninterestingsubjectthatrequiresfurther in-vestigation.

Acknowledgements

Thisresearch was partlyfunded by the ExxonMobilUpstream Researchgrant(EM09296).WewishtothanktheFrenchEmbassy in the USA forthe Chateaubriand Fellowship aswell as the Na-tional Science Foundation Partnership for International Research andEducation(PIRE)grant(NSFOISE-0968313)forpartialsupport. Some of the computational time was provided by the Scientific GroupmentCALMIP(projectP1525),thecontributions ofwhichis greatlyappreciated.

References

[1] Balachandar S , Eaton JK . Turbulent dispersed multiphase flow. Ann Rev Fluid Mech 2010;42:111–33 .

[2] Birman V , Martin JE , Meiburg E . The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations. J Fluid Mech 2005;537:125–44 . [3] Benjamin TB . Gravity currents and related phenomena. J Fluid Mech

1968;31(02):209–48 .

[4] Bonnecaze RT , Hallworth MA , Huppert HE , Lister JR . Axisymmetric particle– driven gravity currents. J Fluid Mech 1995;294:93–121 .

[5] Bonnecaze RT , Huppert HE , Lister JR . Particle-driven gravity currents. J Fluid Mech 1993;250:339–69 .

[6] Bonometti T , Balachandar S . Effect of Schmidt number on the structure and propagation of density currents. Theor Comp Fluid Dyn 2008;22(5):341–61 . [7] Cantero MI , García MH , Balachandar S . Effect of particle inertia on

the dynamics of depositional particulate density currents. Comput Geosci 2008;34(10):1307–18 .

[8] Cantero M , Lee JR , Balachandar S , Garcia M . On the front velocity of gravity currents. J Fluid Mech 2007;586:1–39 .

[9] Cortese T , Balachandar S . High performance spectral simulation of turbulent flows in massively parallel machines with distributed memory. Intl J Super- comput Appl 1995;9:187–204 .

[10] Dade WB , Huppert HE . Predicting the geometry of channelized deep-sea tur- bidites. Geology 1994;22(7):645–8 .

[11] Dade WB , Huppert HE . A box model for non-entraining, suspension-driven gravity surges on horizontal surfaces. Sedimentology 1995;42:453–71 . [12] Fannelop TK , Waldman GD . Dynamics of oil slicks. AIAA J 1972;10(4):506–10 . [13] Felix M , Peakall J . Transformation of debris flows into turbidity cur-

rents: mechanisms inferred from laboratory experiments. Sedimentology 2006;53(1):107–23 .

[14] Fisher RV . Features of coarse-grained, high-concentration fluids and their de- posits. J Sediment Res 1971;41(4) .

[15] Härtel C , Fredrik C , Mattias T . Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. J Fluid Mech 20 0 0;418:213–29 .

[16] Huppert HE , Simpson J . The slumping of gravity currents. J Fluid Mech 1980;99:785–99 .

[17] Hoult DP . Oil spreading on the sea. Ann Rev Fluid Mech 1972;4:341–68 . [18] Kuenen PH . Experimental turbidite lamination in a circular flume. J Geol

1966;74:523–45 .

[19] Lowe DR . Sediment gravity flows: their classification and some problems of application to natural flows and deposits. SEPM Spec Pub 1979;27:75–82 . [20] Maxworthy T . The dynamics of sedimenting surface gravity currents. J Fluid

Mech 1999;392:27–44 .

[21] Meiburg E , Kneller B . Turbidity currents and their deposits. Ann Rev Fluid Mech 2010;42:135–56 .

[22] Middleton GV . Experiments on density and turbidity currents: I. Motion of the head. Canad J Earth Sci 1966;3(4):523–46 .

[23] Necker F , Härtel C , Kleiser L , Meiburg E . Mixing and dissipation in particle– driven gravity currents. J Fluid Mech 2005;545:339–72 .

[24] Parker G , Fukushima Y , Pantin HM . Self-accelerating turbidity currents. J Fluid Mech 1986;171:145–81 .

[25] Ross AN , Linden PF , Dalziel SB . A study of three-dimensional gravity currents on a uniform slope. J Fluid Mech 2002;453:239–61 .

[26] Rottman JW , Simpson JE . Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J Fluid Mech 1983;135:95–110 . [27] Simpson JE . Effects of the lower boundary on the head of a gravity current. J

Fluid Mech 1972;53:759–68 .

[28] Ungarish M . An Introduction to Gravity Currents and Intrusions. CRC Press; 2009 .

[29] Zgheib N , Bonometti T , Balachandar S . Dynamics of non-circular finite-release gravity currents. J Fluid Mech 2015;783:344–78 .

[30] Zgheib N , Bonometti T , Balachandar S . Propagation and deposition of non-cir- cular finite release particle-laden currents. Phys Fluids 2015;27(8):086604 . [31] Zgheib N , Ooi A , Balachandar S . Front dynamics and entrainment of fi-

nite circular gravity currents on an unbounded uniform slope. J Fluid Mech 2016;801:322–52 .