Drag modulation in turbulent boundary layers subject to different bubble injection strategies

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Drag modulation in turbulent boundary layers subject

to different bubble injection strategies

Subhandu Rawat, Agathe Chouippe, Rémi Zamansky, Dominique Legendre,

Éric Climent

To cite this version:

Subhandu Rawat, Agathe Chouippe, Rémi Zamansky, Dominique Legendre, Éric Climent. Drag

modulation in turbulent boundary layers subject to different bubble injection strategies. Computers

and Fluids, Elsevier, 2019, 178, pp.73-87. �10.1016/j.compfluid.2018.09.011�. �hal-02049358�

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Rawat, Subhandu and Ramansky, Rémi, and

Chouippe, Agathe and Legendre, Dominique and Climent, Eric : Drag

modulation in turbulent boundary layers subject to different bubble

injection strategies (2019), Computers and Fluids, vol. 178,, pp.73 - 87

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Drag

modulation

in

turbulent

boundary

layers

subject

to

different

bubble

injection

strategies

Subhandu

Rawat

a

_,

_Agathe

_Chouippe

a,b

_,

_Rémi

_Zamansky

a

_,

_Dominique

_Legendre

a,∗

_,

Eric

Climent

a

a Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse, CNRS, Toulouse, FRANCE b Institute for Hydromechanics, Karlsruhe Institute of Technology, Karlsruhe, 76131, Germany

Keywords:

Turbulent boundary layer Bubbles

Drag reduction

a

b

s

t

r

a

c

t

Theaimofthisstudyistoinvestigatenumericallytheinteractionbetweenadispersedphasecomposed ofmicro-bubblesand aturbulentboundarylayerflow.We usetheEuler–Lagrange approachbasedon DirectNumericalSimulationofthecontinuous phaseflowequations andaLagrangiantrackingforthe dispersedphase.TheSyntheticEddyMethod(SEM)isusedtogeneratetheinletboundaryconditionfor thesimulationoftheturbulentboundarylayer.Each bubbletrajectoryiscalculated byintegratingthe forcebalanceequationaccountingforbuoyancy,drag,added-mass,pressuregradient,andtheliftforces. Thenumericalmethodaccountsforthefeedbackeffectofthedispersedbubblesonthe carryingflow. Ourapproachis basedonlocal volume averageof the two-phaseNavier–Stokesequations. Local and temporalvariationsofthebubbleconcentrationandmomentumsourcetermsareaccountedforinmass andmomentumbalanceequations.Tostudythemechanismsimpliedinthemodulationoftheturbulent wallstructuresbythedispersedphase,wefirstconsidersimulationsoftheminimalflowunitladenwith bubbles.Weobservethatthebubbleeffectinbothmassandmomentumequationsplaysaleadingrole inthemodificationoftheflowstructuresinthenearwalllayer,whichinreturngeneratesasignificant increaseofbubblevolumefractionnearthewall.Basedonthesefindings,wediscussedtheinfluenceof bubbleinjectionmethodsonthemodulationofthewallshearstressofaturbulentboundarylayerona flatplate.Evenforarelativelysmallbubblevolumefractioninjectedinthenearwallregion,weobserved amodulationintheflowdynamicsaswellasareductionoftheskinfriction.

1. Introduction

Itisawell-establishedfactthatanyobjecthavingnonzero rel-ativevelocitywithrespecttotheupstreamfluidsexperiences vis-cousdragatlargeReynoldsnumberduetothethinboundarylayer developingontheobject’ssurface,wherethevelocityofthefluid gradually reducesto that of the objectsurface. The reduction of viscousdragisoneofthemostactiveareasofresearchinfluid me-chanicsbyvirtueofitsimportanceinnumeroustechnological ap-plications, suchasship locomotion,aircraftflight,fuel transporta-tion through pipelines and vehicleaerodynamics. Plethora of ex-perimentalandnumericalstudieshavebeenconductedinorderto manipulateshearinduceddrag.Inthepresentstudywehaveused directnumericalsimulationstostudythedragmodificationcaused by theinteractionofmicro-bubbleswiththenear-wall structures. Ever since thefirst DNSwas conductedby [26],it hasbecome a

∗ _{Corresponding author.}

E-mail address: dominique.legendre@imft.fr (D. Legendre).

fundamentaltooltoabetterunderstandingofturbulence.Withthe rapidadvancementindatastoragecapabilityandprocessorspeed researcherscandoDNSwithalargenumberofbubbles.Whether theresearchesuseexperimentsorDNS,mostofthedragreduction techniquesfocusonthemanipulation ofnear-wallcycleof turbu-lenceregenerationusingeitheractiveorpassivemechanisms.The near-wall structures enhance themixing andthusthe transferof momentumtowardsthewall,hencearedirectlyresponsibleforthe productionofdrag.

1.1. Near-wallregenerationcycle

The flowvisualizationexperimentsperformedby[27]revealed structuresofnear-wallturbulencecalledstreaks.Streaksare quasi-streamwiseregions wherethevelocityislessthanthelocalmean velocity at thesame height fromthe wall. The averagespanwise streakspacing intheviscous sublayer andthe bufferlayer scales inwall unitsandcorresponds approximatelyto

λ

+

z =100,inwall

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shear flows.Numerousexperimental andnumericalinvestigations observed that thesestructures are self-sustained through a cycle ofbreakingupandregeneration.Theflowstructuresarerandomly locatedinspaceandtime,makingverydifficulttoextractasimple mechanism to explain the breakdown andregeneration ofthem. Withtheminimal-flowstrategyof[24]thisrandomnessisreduced and it becomes possible to identify a simple regeneration cycle. Following thisapproach [16] investigatedthe minimumReynolds number and domain size needed to sustain a turbulent regime. The flow visualization andcareful analysis of the energy associ-ated with various Fourier modesevidenced a three-step process. First,theinteractionofthespanwisemodulationofthestreamwise velocity component (i.e.,nearwall streaks) withthe meanshear, leads tothelift-up effect.Thislift-upeffectisresponsible forthe redistribution ofstreamwise momentumthrough counter-rotating quasistreamwise vortices. This redistribution of momentum leads to thetransientgrowthofthe highspeed andlow speed stream-wise streaks. Second, when these streaks reach high amplitude they become unstable to secondary perturbation via inflectional type instability. Finally, the unstable modes of the streaks self-interactto re-energizethestreamwisevortices. Thisphenomenon of lift up andbreakdown is called bursting.[27] reported that a strongfavorablepressuregradientsuppressestheburstingprocess andflow returntolaminarstate. Thisgavestriking evidencethat burstingplaysacrucialroleinturbulencegenerationnearthewall.

1.2. Mechanismofdragreduction

Bursting of near-wall coherent structures intermittently pro-duce andeject turbulent fluctuations intothe core region ofthe domain(see [17]).Inbetweenthetwoburststhereisa quiescent statewherethestreaksgetproduced,becomeunstableand break-down. Numerousattemptshavebeenmadetoreduce theviscous drag under thepremise that thebursting cyclecanbe controlled bythemodificationandalterationofnear-wallcoherentstructures. It isobservedthat streamwiseribletsreduce thedragby stabiliz-ing the coherent streaks (see [4]). [49] used specific patterns of protrusionsdispersedonthewall,whichinterferewiththe coher-entstructuressuchthatitreducestherateatwhichenergyis dis-sipated. Theyfound that thestreakspacingincreasedby 10% and fluctuations of wall-normal velocity component is diminished as compared tothesmoothwall whichwas consistentwithdrag re-ductionusingpolymeradditivesclosetothewall[55].used trans-versetravelingwavesintheirdirectnumericalsimulationsto sup-presswallturbulence.Theyusedfinelytunedtransverseforce con-finedtotheviscoussublayertoinduceatravelingwavewhich con-sequentlyeliminatesthenear-wallstreaks[12].inducedstreaksby generatingoptimalvorticesusingequallyspacedcylindrical rough-ness elementsjustbeforetheleading edgeoftheboundarylayer. Itwasshownthatthestreakssuppresstheinstabilityof Tollmien-Schlichting (TS) waves thereby delaying the transition to turbu-lenceandreducingviscousdrag.

Injections of micro-bubbles ina turbulent flow is another ro-bust method for drag reduction. [38] were the first to demon-strate this phenomenon for fully submerged asymmetric body, wherehydrogenbubblesgeneratedviaelectrolysisintheboundary layer resulted in significant drag reduction. Sincethen numerous approaches both experimental [13,14,18,33,42,45] and numerical [9,11,25,36,44,52,54,54] have been carried out in order to under-standthisphenomenonandtherelativeimportanceofother phys-icalparameterssuchasdiameterandconcentrationofthebubble, bubble deformation, bubble coalescence and breakdown and the methodofbubbleinjectionintotheflow,andpromisingefficiency in actual shipscan now be obtained [28]. [54] numerically sim-ulated afullydevelopedturbulent channel flowatReτ=135and

injected bubblesofdifferentsizes toachievedrag reduction.They

proposed three possible mechanisms: first associated withinitial seeding ofthebubbles, second isrelatedto thedensityvariation which reduces the turbulence momentum transfer and third is regulatedby thecorrelationbetweenthebubblesandturbulence. Theyemphasizedthat unlikethe drag reductiontechniquesof ri-blets, spanwise travelingwaves, ”V” shaped protrusions or poly-merswherethespanwisespacingwasincreasedandstreakswere stabilized,herethestreakspacingseemstobeaffectedvery little andtheybecomemoredistortedduetotheunsteadybubble forc-ing. [9]numerically simulated turbulence boundary layer seeded with micro-bubbles,they took the concentration of bubbles into accountwhilesolvingthecontinuityandmomentumequationsfor the carrier fluid flow. It was reported that micro-bubbles create positive divergence for the fluid and shifts the quasi-streamwise vorticalstructuresawayfromthewalltherebydisplacingReynolds stress

h

u

v

i

peakinpositivewallnormaldirection whichresultsin

thereductionofproductionofturbulentkineticenergy.The span-wise gapbetween thestreaks close to thewall was found tobe increased, this effect was similar to the drag reduction method ofriblets, travelingwavesorpolymer but differentfromfindings of[54].Furthermore,[18]conductedexperimentalinvestigationof theturbulentboundarylayeroveraflatplate with”plateontop” configuration injecting bubbles at the leading edge. They found that bubblesget preferentially accumulatedinthe narrowspatial region close to the wall and the bubble concentration peaks at

y+₌₂₅_due_to_buoyancy_force _that_is_acting_towards_the_wall_for

thisconfiguration.It was assertedthis highbubbleconcentration closeto thewall decreases the coherence ofthenear-wall struc-turestherebyreducingtheirlengthscales.Thiseventuallyleadsto thereductionofmomentumflux towardsthewallwhichresulted inthedecreaseinwallshearstressby25%.

The case of the vertical channel flow has been investigated experimentally [48] and numerically [41], and an increase (re-spectively decrease) of the wall shear stress has been observed inupflow(respectively downflow)configurations.As describedby [41]the presenceof the dispersed phasewill induce,through the forcingterminthemomentum equation,anequivalentadditional pressuregradientequal to±

φ

v

(

andg thegravitationalacceleration) that contributesto a signifi-cantamountofwallshearstressmodification.

1.3.Goalofthepresentstudy

Asdescribedinthesection§1.2significantamountofefforthas beenputinordertoadvancethedragreductiontechniquesinthe lastthreedecades.However, itishardtoignore thatthereisstill noconsensus on themechanism behinddrag reductionobserved duetobubbleinjection.Forinstance,somestudiessuggestthe de-creaseinthecoherence ofthenear-wall structuresduetothe in-teractionofbubbleswithnear-wallturbulence,whereasothers re-portednear-wallcoherentstructures becomingmoreorganizedas aconsequenceofstreaksstabilization.Themechanismresponsible forthe drag reduction is dependent on the voidfraction, bubble diameter,physical forces like buoyancyand liftforces andinitial seedingofbubbles.Thegoalofourstudyistoinvestigate numer-icallythedispersionofbubbles inturbulentboundarylayerflows andtheeffectsofdifferentbubbleinjectionmethods onthedrag modulation. Forthis purpose we first consider the minimal-flow unitconfigurationinordertoisolatetheinfluenceofthedifferent contributionsofthetwo-waycoupling,namelytheforcing dueto momentum exchange betweenthe bubbles and the flow, the in-fluence ofthe localconcentration inthe continuityequation (i.e. divergence effect) and in the momentum equation. We propose fourconfigurationsinwhichthosecontributionsareactivated sep-arately.Then the SyntheticEddyMethod(SEM) isused to

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gener-atethe inletboundary condition forthesimulation ofthe turbu-lentboundarylayers.Anumericalstrategybasedontwodifferent domains is proposed to save computational time. Different bub-ble injection methods for a fixed bulk void fraction were tested toknowwhethertheyhavedifferentimpactontheflow modifica-tionandtheirinfluenceonthenear-wallregenerationcycle. Non-deformablesphericalbubbleswhicharetypicallysmallerthanthe smallest structure of the background flow are considered in this study.Lagrangian trackingofthedispersedphasefits thistypeof physicalconfiguration.Ourapproach isinthiswaysimilar tothe Lagrangian tracking employed by [53] in Taylor-Couetteflow and canreproduceeffectivecompressibilityeffectsduetovoidfraction variationsasdescribedintheworkof[9].

The paper is organized as follows: Section 2 is devoted to a description of the numerical method. We introduce the numeri-cal strategy basedon theSyntheticEddyMethod (SEM)to simu-lateturbulentboundarylayerandwedetailthemodelaccounting for the feedback effect of the bubbles on the carrying phase. In Section3some directnumericalsimulationsofminimalflowunit configuration were conducted to study the preferential accumu-lationof bubblesandinvestigatetheir influence onthe turbulent structures andregenerationcycle.Finally,themain resultsonthe dragmodulationrelatedtotheinjectionofbubblesinaboundary layerarereportedinSection4.MorespecificdetailsontheSEMis providedinAppendixBfollowedbyavalidationofthesimulation oftheturbulentboundarylayeraswellasasetofverificationsof theimplementationofthe differentcontributionsofthetwo-way couplingofthebubblephaseontheliquid.

2. Numericalmethod

ThenumericalapproachisbasedonEuler–Lagrange modeling: the continuous phase flow is predicted through direct solution of the Navier-Stokes equations while Lagrangian bubble trajecto-riesarecomputedbynumericalintegrationofmomentumbalance equation. Modeling the presence ofbubbles isbased on volume-averaging of the momentum and continuity equations including momentumsource termstogetherwithspatialandtemporalvoid fractionvariations.

2.1. Equationforthecarryingsinglephaseflow

Assuming constant physical properties (viscosity and density) for a Newtonian incompressible fluid, we consider the Navier– Stokesequationsforthecontinuous(liquid)phase

∇

·u=0 (1)

∂

u

∂

Thesystemofequationsisdiscretized onastaggerednonuniform grid witha finite volume approach.Spatial derivatives are calcu-latedwithsecondorderaccuracyandweusesemi-implicit Crank-NicolsonschemefortheviscoustermandthreestepsRunge–Kutta scheme for time integration. The JADIM code has already been widelyusedandvalidatedforlaminarandturbulentconfigurations insingle-phaseconfigurations(seeforexampleduringthelasttwo decades[2,15,31,35,40]).

Spatially developing flows asa turbulent boundary layerpose some challenges:if one wants to avoid an unnecessary long do-main, time-dependentinletboundary conditionsneed tobe con-sidered. The flow downstream is highly dependent on the inlet condition whichshould bevery closetotherealturbulence. Tur-bulent inlet generation techniques for spatially developing flows havebeenan active areaofresearch fromthelast threedecades.

Forbubbleladenturbulentboundarylayer,[10]adaptedaspectral method to synthetically generate turbulent initial condition with prescribed energy spectra and Reynolds stress tensor. With this initialization theydida smallprecursor simulationandcombined it withthemethodof[32] to generatetheinletvelocity field for themainsimulationwithlongerdomain.Inthepresent studywe implemented SyntheticEddy Method(SEM) proposed by [20] to generate theinlet boundary condition. SEMgivesa time depen-denttwodimensionalvelocityfieldattheinletplanewiththe pre-scribedmeanandcovariancealongwithtwopointsandtwotimes correlations. Thisapproachalongwiththenumericalimplantation details arepresentedinAppendixA.Notethattherecycling tech-nics([51],[50],[32]) arenotconsidered hereduetothetwo-way couplingwiththebubblyphase.Indeed,theflowisalready mod-ifiedandthereforecannot be recycledastheflow isnot pseudo-periodicanymore.

2.2. Lagrangiantrackingofbubbles

Weutilizetwo-waycoupledEuler-Lagrangianframeworkwhere the dispersed phase is composed of N_b pointwisespherical bub-bles. Bubble trajectories are computed solving Newton’s second law,wheretherateofchangeofmomentumofthebubbleisequal to thetotal external forces

6

F_b actingon it. The external forces consist ofdrag,fluid acceleration,added-massandliftforces.The radius, volume andvelocity of the bubble are noted asRb, Vb=

(

4/3

)

π

R3

b andv,respectively.The vorticityoftheliquidis

Ä

.The

bubbleforcebalancecanbewrittenas:

dx dt =v (3)

ρ

bVb dv dt =

(

v−u

|

2 d/

σ

based

ontheslipvelocity

u−v

|

2Rb

ν

, (6)

theadded-masscoefficientisCM=1/2andtheliftcoefficientCL is

givenbytherelationof[31]: CL=

·

1.8802

(

1+0.2Reb/Sr

)

3 1 RebSr +

³

₁ 2 1+16/Reb 1+29/Reb

´

2

¸

1/2 (7)

withSrthelocalshearratenondimensionalizedbytherelative ve-locityandthebubblediameter.

Directbubble/bubbleinteractionsareneglectedconsideringthat the averaged void fraction in Section 3 and 4 is

8

v=0.02 and 0.001, respectively. However,the effectofconcentration peakson bubble/bubble interaction will be discussed in Section 4.1. Bub-ble/walloverlapispreventedbyassuminganelasticbouncingwith theplanewall.Thecomputationoftheseforcesrequiresthe inter-polationofthefluidvelocity anditstimeandspacederivativesat

c

locally whichisunity ifthereis fluidatlocation xattime t and zerootherwise.

Then thelocalvolumefractionofthecontinuousphase

ε

c(x,t)

ε

c

(

x,t

)

= 1

V ZZZ

V

χ

c

(

ξ

,t

)

d3

ξ

=

h

χ

c

i

(

∇

Uc

)

T

¤

] +Fd→c (10) Fd→c=− 1 V Nb,l,(i, j,k) X b=1

· ρ

bVb

(

dvb dt −g

)

¸

. (11)

Theinfluenceofthebubblesonthecontinuousphaseisrelatedto thelocalforcinginducedbyeachbubbleviathemomentumsource termFd→c andtothelocalevolutionofthefluidvolume fraction

ε

c.Inthoseequations,wehaveneglectedmomentumtransportby

small-scale fluctuations(scales smaller than the control volume). The corresponding tensor

h

u′

iu′j

i

c, similar to the Reynolds stress

tensor,mayhavetwomainphysicalorigins:theclassicalturbulent fluctuation ofthecontinuousphase,andthebubbleinduced fluc-tuationsduetofinitesizeeffectsandwakeinteractions([47]).The formercontributionisnot consideredsince weintendtoperform directnumericalsimulationsofthecontinuousliquidflowwhileit isreasonabletoneglectthelatterwhenconsideringdilutebubbly flows.Notethat,insomestudies(seeforexample[5,9,29,37]),the fluid inertia is subtracted from(11), becauseinthese studiesthe volume occupiedby the bubbles in the cell is not considered in themomentumbalanceequation(10)through

ε

c(x,t).

Fig. 1. Schematic representation of bubble close to the wall. Cell 4 contains the center of the bubble, the contribution due to this bubble to εc is accounted at cell

4 only. Cell 1, 2 and 3 also share some portion of this bubble therefore we set

εc(1) = εc(2) = εc(3) = εc(4) .

The system ofequation is discretized on a staggered nonuni-formgridwithafinitevolumeapproach.Thepressureandthe liq-uidvolumefractionarelocatedatthesamenodeandthevelocities arefacecentered.In thecell(i,j,k)ofvolumeV_{i, j,k},the valueof

ε

c,(i,j,k)isdirectlycalculatedfromthenumberofbubblesNb,(i,j,k)

presentinthecell(i,j,k)by:

ε

c,(i, j,k)=1−

ε

d,(i, j,k)=1− Nb,(i, j,k) X b=1 V_b V_{i, j,k} (12)

The point particle approach is well suited when the size of thebubbles is smallerthan thesmallest scaleof variationof the fluidflow.Fig.1givesaschematicrepresentationofasingle bub-bleclosetothewall.Gridsizeinstreamwiseandspanwise direc-tionsis muchlarger thanthebubblediameter.Eq.12 isvalidfor

R_b≪

1

,butclosetothewall

1

y>Rb.Althoughbeyondthemodel’s

assumptions,itisunavoidable tosimulatelarge Reynoldsnumber flowswithintermediatebubblesizeatareasonablecomputational cost.Theminimumpossibledistanceofthebubbleisy+₌_R+

b from

thewall.Thevalueof

ε

cisbasedonthepositionofthecenterof

the bubble,as a consequence,the cells closeto the wall (where

Rb>y)doesnotfeelthepresenceofthebubbles(

ε

c=0).To

rem-edythisproblemandpreventthelargewall-normalgradientof

ε

c

inthe vicinity ofthewall we impose a constant value for

ε

c for

thecells locatedbetweeny=0 andy=Rb.We note thatthe

ap-proximationisrathercrudebutwetestedthatwithoutanyspecific treatment forthe value of

ε

c closeto the wall and we obtained

qualitativelysimilarresults.

In Appendix B we provide various verification and validation of the numerical development proposed for this paper. Namely, B.1 presents validation of the single phase turbulent boundary layersimulations against the numericalsimulations of[23].Then inB.2andB.3,weproposesimpleoriginaltest casestocheck the two-waycouplingmethodin orderto verifyeachcontribution of the terms induced by the presence of bubbles in the system of equations(9–10).

3. Minimal-flow-unitconfiguration

As seen fromthe Eqs. 9 and 10, the presence of bubbles af-fectstheflow through differentcontributions.There isa void ef-fectinbothmomentum andcontinuityequationsaswellasa di-rectcontributiontothemomentumcoupling(Eq.11).Thepointof thesectionistostudytheeffectofeachofthesecontributions(as well astheirpossible interplays)on thenearwall flow structure. We considerin thissection the minimal flow unit [24] in which adynamicalprocessforthegenerationofthewallflowstructures isextracted.Morespecifically,tostudytheinteractions ofthe dis-persedbubblyphasewiththeself-sustainedwallstructure mecha-nism,westudythefollowing5cases:

mfu0:One-waycoupledsimulationi.e. Fd→c=0 ineq 10and

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Fig. 2. (a) mean bubble concentration profiles ( εd ( y )) as a function of wall normal coordinates for mfu0,1,2,3,4 . (b) mean lift, drag and added mass forces acting on the

bubbles for mfu0 (black) and mfu4 (red) normalized by the friction velocity of the mfu0 case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1

Computational domain and discretization parameters used in the minimal-flow- unit box at Re τ= 180 .

Name L x L y L z N x N y N z S t 8v R +_b Re τ

mfu0 2.2h 2h 1h 32 82 24 0.28 0.02 1.3 180

mfu1: Only momentum feedback from the dispersed to the fluid phase isconsidered i.e. Fd→c=6 0 ineq 10 and

ε

c=1

and

τ

∗=

ν

/u2τ is the time scale based on

friction velocity uτ.We useperiodic boundaryconditions forthe

velocity in all three directions, and a mean pressure gradient is imposedinordertoobtainedtheprescribedflowrate.Pleasenote that the caseofthemfu4 simulationshave alsobeenperformed withatwice finermeshinall directions,andthatnosizable dif-ferenceswereobserved.Thereforeweconcludethatthemeshused forthesimulations describedinthepaperis fineenough to cap-tureallthehydrodynamicfluctuations.

In case of one-way coupled minimal-flow-unit (mfu0) the streamwiseandthespanwisedimensionsofthecomputational do-main are L+

x ∼400 and L+z ∼180 respectively in wall units. The

Reynolds number basedon uτ corresponds to Reτ∼180.Spacing

betweenthenear-wallstructuresisfoundtobearound

λ

+

z ∼80−

100 thereforemfu0could have two pairsof highandlow speed streaks.Thissizeischosensothatbubbledynamicscouldbe stud-iedduringtheinteractionofthesestreaks.Wesetthebubble vol-umefractionat

8

v=0.02alongwithabubbleradiusofR+_b ≈1.3, correspondingtoaStokesnumberSt≈0.28.

In Fig.2(a) we present themean bubbleconcentration profile asa function of the wall normalcoordinate. The bubble concen-tration is computedby averaging over time anddirections paral-leltothewall.Weremarkthattheprofileobtainedformfu0and

mfu1 are very close and decrease towards the wall whereas in themfu2,3,4thereisasharpincreaseinthebubble concentra-tioncloseto thewall.Sincethedifferentbubbledistributions are

only dueto a modification ofthe carrierflow, we concludethat, atthisvolumefraction,voideffectsinboththecontinuityor mo-mentumequationshavemuchmorepronouncedeffectontheflow thandirectinterphasemomentumcoupling.Toanalyzefurtherthe strong modificationofthe meanbubbleconcentration we givein figure 2(b) the mean drag, lift and added mass forces actingon thebubbleasafunctionofwallnormaldistanceforthemfu0and

mfu4. It appears that inmfu4,lift anddragforces are modified when the voideffectalters thecarrierphase flow andthere isa negativepeakinthedragforceclosetothewall.Asaconsequence, thereisanetforce actingonthebubblestowardsthewallwhich resultsinhighbubbleconcentrationclosetothewall.

The second-order turbulent statistics are analyzed to further understand the effectof bubbles on the fluid phase. In Fig. 3(a) root-mean-squarevelocity inthe streamwiseandwallnormal di-rectionisshownasafunctionofwalldistanceforall cases.Inall casesexceptmfu1turbulenceintensityishigherthansinglephase case and the peak is slightly shifted towards the wall. Similarly, Reynoldsstressprofileisalsomodifiedduetothepresenceofthe bubble, higherturbulent stress intensityis observed inmfu2,3,4

ascomparedtomfu0andmfu1(Fig.3(b)).Thepeak ofReynolds stress ismoved towardsthewall whichinturnincreasesthe pro-ductionofturbulentkineticenergyproductionnearthewall.

Reynoldsstressisdeterminedbythecontributionofthe burst-ing andsweepevents,thereforewe investigatetheeffectof bub-bleinjectionsontheburstingandsweepcycle.Contributiontothe Reynolds stress from the fourquadrants (Fig.4) iscomputed for

mfu0 and mfu4(see Section 4.4 for details). Negative contribu-tiontotheReynoldsstress(-u

v

)fromthesecondandfourth

quad-rant accounts for the burst andsweep events.Close to the wall sweepeventdominatesandburstingismoreprominentawayfrom the wall.Thecrossovertakesplaceaty+_≈₁₅_for_mfu0_._In_mfu4

wheredragincreasewas observed,thenegativecontributionfrom both sweepandbursteventshaveincreasedandthepointwhere they first cross each other is shifted towards the wall (y+_≈₁₄_).

The positivecontributionto-u

v

fromthefirst andthirdquadrant

isreducedinmfu4ascomparedtomfu0.

TheabovetestcasesindicatedthatfeedbackforceFd→c6=0has

minimal effectonthenear-wallstructureordragmodulation.The mostimportantfactoristhevoideffectinthecontinuityand mo-mentumtransferequations,whichplaysacrucialroleinthe mod-ification ofthenear-wall coherentstructures andsignificantly in-creasesthebubbleconcentration nearthewall.Dragchangeis al-most samein both mfu2and mfu4asreported inTable 2. This similarity between mfu2 and mfu4 is also clear from rms and Reynolds stress (-u

v

) profiles inFig.3. From theabove test cases

itisclearthatbubbleconcentrationclosetothewallhasa domi-nanteffectonthewallfriction.

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Fig. 3. (a) Comparison of u rms(y +) and v_rms(y +₎_{for the}_mfu0,_1,_2,_3,₄_{cases. Each curve is normalized by the friction velocity of the}_mfu0_{case. (b) Reynolds stress profiles}

−u v normalized by the friction velocity of the mfu0 case.

Fig. 4. Contribution to the Reynolds stress from the four quadrants. black lines:

mfu0 and red lines: mfu4 . Each curve is normalized by the friction velocity of the

mfu0 case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2

Drag modulation in different cases.

Name mfu0 mfu1 mfu2 mfu3 mfu4

Cf−Cf,0

Cf,0 0% 0% 8.5% 10.3% 8.5%

4. Modificationofaturbulentboundarylayerduetobubble injection

Wenowconsiderthecaseoftheturbulentboundarylayer.The flow is generatedwiththe aid ofthe SyntheticEddyMethod in-troducedinSection2anddetailedinAppendixA.Table3 summa-rizesthespecificitiesofthecomputationaldomainand discretiza-tionusedinthecurrentsection.Formoredetailsonthevalidation ofthesimulationofthesingle-phaseturbulentboundarylayerthe readerisreferredtoB.1.

Table 3

Computational domains and discretization parameters used in turbulent boundary layer simulation.

L x L y L z N x N y N z 1+x y +min 1+z

10 δ0 2 δ0 3.56 δ0 360 66 136 18 0.14 9.0

4.1. Influenceofthebubbleinjectionstrategy

Contrary tothe minimal flow unit considered inthe previous section,intherealwallflowsatlargeReynoldsnumber,thereare multiplenear-wallstructuresalongwiththelarge-scalestructures. [36]intheir DNSofturbulentboundarylayerwithmicro-bubbles showed that bubbles migrate away from the wall as they move alongthe streamwisedirection andthisdecreases their abilityto modulatethenear-wallcoherentstructures.InTurbulentBoundary Layer(TBL)simulationof[9],anytimeabubbleleavesthedomain itisrandomlyreinjectedinsidetheturbulentboundarylayer.With thisinjectionprocedure,thebubbleconcentrationprofilespeak af-tery+₌₁₀₀_and_when_the_average_volume_fraction_of_the_bubbles

islow (

ε

d=0.001) thereare veryfew bubblesnearthewall (see

Fig.5).Forthisreason,themodificationofthedragcoefficientand thenear-walldynamicsis minimal.Therefore,higherbulk bubble volumefractionisrequiredtogeneratesomesignificantchangesin drag.Thealternativetoenhancethebubbleinteractionwith near-wall coherent structures for a fixed global volume fraction is to injectbubblesatdifferentheightsclosetothewall.

We considerfive casesof bubbleinjections assummarized in Table4. Inthe first fourcases (denoted:2Rb, 4Rb, 6Rb and8Rb)

bubblesare injected closetothe wallatdifferenty+_. _In_the_fifth

case(named:Random),thebubblesare spontaneouslynucleated atrandompositionsinthewholedomainwithavelocityequalto thelocalfluidvelocity.Inadditionthosesimulationsarecompared

Fig. 5. Panel (a) on the left compares the mean bubble concentration profiles, panel (b) shows the streamwise evolution of the friction coefficient C f ( x ), the solid black line

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Fig. 6. Comparison of the velocity RMS for Single-phase , 2Rb , 4Rb , 6Rb and 8Rb cases. (a) variance of the streamwise u rms and (b) wall-normal vrms velocity components

normalized by the free stream velocity U 0 as a function of wall-normal coordinate ( y ). Color coding is similar to Fig. 7 .

witha singlephase flow whoseparameters are givenin Table3. Inall casestheoverall bubblevolumefractionwithin the bound-arylayeriskeptfixedat

8

v=0.001.Notethatinthecasedenoted 2Rbthebubbleradiusisonefourthoftheradiusoftheotherthree

cases,thiswasdone toinvestigatetheeffectofinjectingthe bub-blesveryclosetothewall(the injectionpositionhastobelarger than the bubbleradius). However, note that since the bubble di-ameterisalreadysmallerthantheturbulentwallstructureswedo notexpect averyimportanteffectofthebubblediameteraslong asthegasvolume fractionisconstant.The caseRandom is sim-ilar to the case-Bof [9] in which whenevera bubbleleaves the computationaldomainitisreinjected atarandom locationinside theboundarylayer.Inothercaseswhenthebubblesleavethe do-main they are reinjected at random positions on the x−z plane

located atthe height of2Rb, 4Rb, 6Rb and8Rb from thewall

re-spectively. Note that forthe bubble diameter considered here, it can be checked that for air bubbles in water and for a realistic boundarylayerthicknesstheWebernumberisactuallyvanishingly small,consistentlywiththeassumption ofsphericalbubbles con-sideredinSection2.2.

Figure 5(a) presents theprofile of theaverage bubble volume fractionofthevariouscases.Theprofilesarecomputedfrom spa-tialaverageinspanwisedirectionandtimeaveragingandaregiven for a streamwise location x=10

δ

0, for which Reθ≈1100. Bubble

concentration profilepeaksatthelocationofthebubbleinjection andthengraduallydecreasesaswemoveawayfromthewall.This explainsthattheconcentrationprofilesdifferfromthetrendofthe literature ([9,36]) withmuchmoreintense peaksthat canbealso locatedinthebufferlayer.Thepeakofbubbleconcentrationinthe nearwall region suggeststheexistence of bubble/bubble interac-tions,whichhavebeenneglectedinthepaper.Todiscussthe im-pactonthebubble dynamicsoftheinducedbubble/bubble inter-actionsinthenearwallregion,weconsiderthebubble-bubble dis-tancecomparedtothecharacteristiclengthoftheturbulent struc-tures.Theminimumaverageinter-bubble distance,inwallunit,is expressedas

δ

+ b =R + b

µ

4

π

3

ε

d

¶

1/3

.Takingthevalueofthelocal con-centrationpeak,

ε

d∼0.1(seefig.10)gives:

δ

b+≈5,whichremains

smallerthanthewallstructuresintheverynear-wallregion.This indicates thatthebubblesinteractmuchmorewiththeturbulent structures thanwithotherbubbles, andthat bubble-bubble inter-actionsarenotexpectedtobedominantinthisregion.

The evolutionofthe temporalandspanwiseaverage skin fric-tion coefficientCf

(

x

)

=2

h

τ

w

i

(

x

)

/

ρ

U2 _in _the_streamwise _direction

isshowninfigure5(b),here

τ

w=

µ

dU/dy.Inthecaseof2Rb and

4Rbthedragissmallerthanintheone-waycoupledcase,forcase

Random the dragseems to be unchanged, whereasthe drag in-creasesin cases6Rb and8Rb.Thisincrease canbe interpretedas

the consequence ofthe wall-normal location of bubble injection.

Notethat thesmallvariationsbetweenthecases2Rb and4Rb

in-dicatethatthemainparameterforthealterationofthewall struc-turesisthelocalvolumefractionofthenearwallregion.The bub-ble diameter and the bubble number density separately are not dominant. Thisisconsistentwiththeprevioussectionwhereitis observed thatthebubblesmainlyinfluencethewallthroughvoid fractioneffectandnotdirectmomentumcoupling.Thisconclusion onlyholdsforsmallenoughbubbles,forlargebubblesthe momen-tumforcingcausedbyeachbubbleshouldprevail.Theinduced ef-fect of the injection position is further detailedin the following consideringtheReynoldsstressanditsanalysisusingquadrant de-composition.

4.2. Turbulentfluctuationsandenergy-Spectrum

The turbulent statistics (urms=

p

u′2₎ _computed _for _the _cases

discussedaboveareshowninFigs.6(a)and(b).Thevelocity com-ponents are normalized by the free stream velocity U0. In cases

2Rb, 4Rb streamwise and wall-normal turbulent energy

fluctua-tionsareslightlylowerthantheSingle-phasecase,whereasthey are higher in 6Rb and 8Rb. These findings are consistent with

observed modification ofthe skin-friction coefficient. Similarly, a reduction of the turbulent fluctuation was reported by [11] and [44] in their DNS of turbulent boundary layer andchannel flow simulations laden with micro-bubbles. However, unlike them we donotobserveanynoticeableshiftinthermspeakalongthe wall-normaldirection.Itispossiblethatbecausethedragreduction ob-served hereislessthan4percent,theshiftinwall-normal direc-tionisnegligible.

To analyze the cause of drag modification due to this injec-tionstrategypre-multipliedspanwiseenergyspectrumisanalyzed forcases2Rb, 4Rb,6Rb,8Rb andSingle-phasecase. Figures7(a)

and7(b)showpre-multipliedspanwiseenergyspectrafor stream-wise (KzEuu) andspanwise velocity components (KzEww) atthree

different heightsfrom the wall (y+₌₄_.₀_,₁₀_,₁₅_). _In _all _the _cases

thespectralenergypeaksatthewavelengthof

λ

+

z ≈80−100and

thispeakshiftsgraduallytowardshigherwavelengthsaswemove awayfromthewall.Thiswavelengthcorrespondstothe character-isticspacingbetweenthe streakscloseto thewall.However, 2Rb

and4RbhavelowerenergycontentthantheSingle−phase

simula-tion,whereas6Rband8Rbhashigherenergycontent.Asexplained

inSection 1.1near-wallcycleconsistsofstreakswhichafter gain-ing sufficientenergy become unstableand breakdown to gener-atevortices.Incases6Rband8Rb higherenergystreaksaremore

pronetoinstabilitywhichresultsintheireventualbreakdowninto vortices. Increaseininstability ofthecoherentstructures leadsto moreviolentburstandsweepevents,whichresultsintheincrease in turbulent shear stress production (see Fig. 9). Opposite could be said aboutthe case 2Rb and4Rb where energy inthe

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coher-Fig. 7. Pre-multiplied spanwise energy spectrum comparison for Single-phase , 2Rb , 4Rb , 6Rb and 8Rb case at y + = 4 . 0 , 10 , 15 . Panel (a) on the left shows pre-multiplied

streamwise energy spectra ( K z E uu ) for streamwise velocity component and Panel (b) on the right shows pre-multiplied spanwise energy spectra ( K z E ww ) for streamwise

velocity component. To better discern the curves at y + = 10 and 15 for K

z E uu we have plotted 5 · 10 3 + K z E uu at y + = 15 . Similarly, 1 · 10 3 + K z E ww is plotted at y + = 15 for

K z E ww .

Fig. 8. Single-phase , 2Rb , 4Rb , 6Rb and 8Rb cases are compared. Panel (a) on the left compares the Reynolds stress profiles −u v as a function of y + , panel (b) on the right

compares turbulent energy production term ( P = −u v d U/d y ) as a function of y + . All curves are normalized with the free stream velocity. Color coding is similar to Fig. 7 .

entstructuresisreducedwhichmakesthemlessunstableand de-creasesturbulentshearstressproduction(seefigure9).

4.3. Shearstressandturbulentenergyproduction

We examine the Reynolds stress (−u

v

) and turbulent energy

productionterm(P=−u

v

dU/dy)inFigs.8(a),(b)respectively.Itis

apparent thatwhencompared toSingle−phasecasetheReynolds stressisreducedfor2Rb,4Rbcasesandremainsalmostunchanged

for Randomcase, whereas for6Rb and 8Rb it increases.Reynolds

stressprofileismildlyshiftedawayfromthewallincases2Rband

4Rb compared to Single−phase case. However, forcases 6Rb and

8Rb theReynolds stress profileshifts towards the wall. Reynolds

stress profile peaks at y+_≈₆₉ _in_Single₋_phase _case, _it _shifts _to

y+_≈₇₃_for_2R

band4Rbandtoy+≈64for6Rband8Rb.Theshift

in Reynoldsstress away fromthewall could be attributedtothe shift invortex cores tohigher y location.The opposite trend oc-curs for6Rband8Rb wheretheReynoldsstress peakismovedto

lower ylocation inferringthemovementofvorticalstructures to-wardsthewall.TurbulentenergyproductionPtermappearstobe decreased forcases 2Rb and 4Rb and increased for6Rb and 8Rb

whencomparedtoSingle−phasecase.

δ

0 for the first fourcases in Table 4.Very close

tothe wall, sweeping events(Q4) is aleading contributorto the turbulent Reynolds stress. For the single phase case, at around

y+_≈₁₄_the _contribution_of _Q2 _becomes_equal _to_Q4 _and _further

awayburstingeventsdominatetheReynoldsstress.Forcases6Rb

and8Rb this intersectionpoint for Q4 andQ2 mildly shifted

to-wardsthewalltoy+_≈₁₃_.₆_&_13.4_{respectively.}_On_the_other_hand,

forcases2Rb,4Rbthisintersectionpointismovedawayfromthe

wall to y+_≈₁₄_.₆ _& _15.11 _{respectively.} _In _cases _6R

b and8Rb the

intensity of sweep and ejection is enhanced as compared to all the other cases. Whereas in cases 2Rb, 4Rb contribution to the

Reynoldsstress dueto sweeping and bursting eventsis reduced. ThepositivecontributiontotheReynoldsstressfromquadrantQ1 andQ3 is enhanced incases 2Rb and4Rb andreducedfor cases

6Rband 8Rb.

Thus,forcases6Rb and8Rb thebubbleinjectionincreasesthe

intensityof burstingand sweeping eventsthereby increasing the Reynoldsshear stress.Whereas, in cases2Rb, 4Rb, thereis a

de-crease of the negative contribution from Q2 and Q4 and an in-creaseof the positive from Q1 and Q3, which results inthe de-crease in total Reynolds stress. As burst andsweeps are located respectively above the low and high speed near-wall streaks i.e

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Fig. 9. Single-phase , 2Rb , 4Rb , 6Rb and 8Rb cases are compared. Panels (I), (II), (III) and (IV) are arranged according to the four quadrants of the u-v plane. Color coding is

similar to Fig. 7 .

Fig. 10. Contour of λ2 < 0 is shown at 10% of its absolute value in light green

color up to streamwise distance of x = 8 δ. Bubbles positions are shown with black spheres. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

adjacent to thequasi-streamise vortices (see [9]) change in their intensitymustbecorrelatedtotheinteractionofbubbleswiththe near-wallcoherentstructures.

4.5. Bubbledispersion:

λ

2criterion

Inthissection,preferentialaccumulationofbubblesinducedby the coherent structures is examined. Bubbles and vortical struc-tures mutually interact with each other in a turbulent flow. Lift force,localpressuregradientandaddedmassforceencouragethe accumulationofbubblesinlowpressureregionsoftheflow[8,37]. Such low pressure regions, which often correspond to the vor-tex cores, are classically identified by

λ

2 criteria ([22]). It

cor-responds to negative values of the second-largest eigenvalue of

S2+

Ä

2whereSand

_Ä

arethesymmetricandantisymmetricparts ofthevelocitygradienttensor

_∇

u.Figure10showsthecontourof

2<0.Fig.11(a)

shows the probabilities Pf

λ2 and P

b

λ2 as a function of wall

nor-mal distance for all the cases listed in Table 4. Basically, for all the case Pf

λ2 presentsa similar profile,with a minimum around

Fig. 11. Probability of occurrence of λ2 < 0 for cases listed in Table 4 and

Single-phase case. P f

λ2(y ) and P λ2b(y ) are shown with lines and points respectively.

Table 4

Bubble radius and distance to the wall of the injection plane for the different run. For all cases the overall bubble volume fraction within the boundary layer is 8v = 0 . 001 . Case 2Rb 4Rb 6Rb 8Rb Random R + b 0.3 1.3 1.3 1.3 1.3 y + in j 0.66 5.28 8.0 10.5

-y+₌₂₀_._More_{specifically,}_for_the_Single-phase_case_Pf

λ2 peaksat

y+_≈₁₉_.₅_, _whereas _in_cases _2R

b, 4Rb, 6Rb and 8Rb thepeak

re-spectivelyshiftstoy+_≈₂₁_.₂_,₂₁_.₅_,₁₆_.₈_and_16.7._For_the_cases_2R b

and4Rbwheredragreductionwasobserved,thepeakofthe

prob-ability distribution Pf

λ2 shifts away from the wall, while in case

6Rband8Rbitslightlyshiftstowardsthewall,andfortheRandom

case no such shiftwas observed. This indicates that the vortical structures aredisplacedtohigherwallnormallocationswhenthe drag isreduced andthey move towards the wall incaseof drag increase.Overall,theprobabilityPb

λ2> P

f

λ2 demonstratesthat

bub-bles in turbulent flow get preferentially accumulated in the vor-texcores,whichisconsistentwiththeobservationsof[6]and[3].

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At thedistanceto thewall corresponding tothe bubbleinjection there isa sudden variationinPb

λ2 butfurtheraway,sayy

+_>₁₀_,

bubbles accumulate in

λ

2<0 regions. Moreover, the probability

distribution Pb

λ2 becomes similar away fromthe wall for all the

casescorrespondingtobubbleinjectionatthewall(2Rb,4Rb,6Rb

and 8Rb). This is in contrast with the Random case which

devi-atesfromtheothercases.Thisisduetospontaneousnucleationof bubblerandomlyinthewholedomain.

5. Conclusion

We develop further the numericaltoolsnecessary to simulate aspatially developingturbulentboundarylayerladenwith micro-bubbles. We have implemented a numerical approach which en-ables tocomputebubbledispersionwithtwo-waycouplingbased onEuler-Lagrangeapproach.Theformulationofthefeedbackofthe dispersed phase on thecontinuous one is basedon local averag-ing thatmodifiestheNavier-Stokes equations.Tostudythe physi-cal mechanisms atplay inthe alterationofthewall flows bythe presence ofmicro-bubbles,we considered twodistinct situations. Firstweconsideredaminimalflowunitladenwithmicro-bubbles. Thismodelflowenablestoisolatetheself-sustainprocessof gen-eration of the wall structures. Our results demonstrate that the bubbles tend to migrate closeto the wall mainly dueto the oc-currence ofsweep events.Thismigrationcanbe attributedtothe relative low Reynolds numbers explored for those configurations andthe simplifieddynamicsof theminimal flowthat might em-phasizetheinfluence ofthose sweepevents.Moreover,wenotice thatthepresenceofthebubblestendstoreinforcetheoccurrence of suchviolent events.Alsoit appears thatthe bubblesinfluence the flow mainlythrough effects causedby thevoid fraction fluc-tuations (both in momentum and continuity equations) and not through direct momentum coupling. Since mostof the effect on thewallshearstressappearstobeduetothebubblesintheclose vicinityofthewall,wediscussedtheinfluenceofbubbleinjection methodsonthemodulationofthewallshearstressofaturbulent boundarylayer.Weobserveachangeintheflowdynamicsaswell as a modification of the skin friction, even fora relatively small bubble volume fractioninjected in thenear wall region.The key parametersseemtobethevoidfractionandtheinjectionposition whereasthebubblesizeislessimportant,atleastaslongas bub-blesaresmallerthanoroftheorderofthefrictionlength.Itseems thatthedragisdecreasedwhenbubblesareinjectedintheviscous sub-layerwhileitisincreasedwhentheyareinjectedinthebuffer layer.Weshowthatthisdragbehaviorisrelatedtoachangeofthe nearwallflowstructures,andthusoftheirregenerationcycle, de-pendingonthebubblesinjectionlocation.Theresultingchangein theintensityoftheso-called”sweep” and”burst” causesa modifi-cationofthemeanstreamwisemomentumfluxinthewallnormal direction,asconfirmedbytheReynoldsshearstressprofiles result-inginthechangeoftheskinfriction.However,itremainsanopen problem inturbulence howto quantitatively relate the alteration oftheregenerationcycle(oratleasttheevolutionofthedifferent quadrantscontribution)totheoverallwallshearstress.

Acknowledgments

We acknowledgethesupportofthefrench ANR(Project num-ber ANR-12-ASTR-0017). This work was performedusing HPC re-sourcesfromGENCI-CINES/IDRISandCALMIPcenterofthe Univer-sity ofToulouse.We alsothankAnnaïgPedrono forher technical support.

AppendixA. TheSyntheticEddyMethod(SEM)

TheSyntheticEddyMethodSEMtechniqueisbasedonthe hy-pothesisthat turbulencecanbemodeled assuperpositionsof co-herent structures. [46] suggested that superpositionof analytical eddieswith shapesinspired by realcoherent structures provides a decent approximationof lower-order statistics ofwall-bounded flows. Therefore, SEM deals with generating synthetic eddies at theinletplane usingspecificshape functionwhichcontainstheir spatial andtemporal characteristics. The inlet plane is located at

x=0andhas physicaldimensions[0, Ly]×[0, Lz]in wall-normal

andspanwisedirections, whilethe full domainhasthe length Lx

inthestreamwisedirection.Thegridisuniforminthestreamwise (x)andspanwise(z)directionswithresolution

1

xand

1

z respec-tivelyandstretchedinwall-normaldirection(y)usingatangential stretchingfunction,thus

1

yincreasesawayfromthewall.

Asafirststepathree-dimensionalvirtualboxBcontaining the eddies is created around the inlet. The dimensions of the box B

are [−lx,lx]×[−ly,Ly+ly]×[−lz,Lz+lz] and eddiesare generated

atsomerandomlychosenlocationsinsidethisdomain.Thevolume oftheboxisdenotedbyVBandthenumberofeddiesproducedis NE.

Thestochastic velocity signal u˜ ata point xis assumedtobe thesuperimposition ofthecontributionfromalltheeddies NE in

theboxB, ˜ ui

(

x

)

= 1

p

NE NE X k=1

ǫ

k_f σ (x)

(

x−xk

)

(A.1)

z−z k

σ

z

´

(A.2)

fhasacompactsupporton[−1,1]andsatisfiesthenormalization condition

Z1

−1

f2

(

x

)

dx=1 (A.3)

Fig. A1. Pictorial representation of synthetic eddy method. Purple square represent the inlet plane where the initial turbulent condition is supposed to be generated. Black domain is the boundary of the virtual box whose size is dependent on the length scales of the eddies generated.

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Fig. A2. Schematic representation of Domain-A - Domain-B configuration.

Inthepresentstudyfischosentobeatentfunction, f

(

x

)

=

½p

)

) andtovary withthe wall normaldirection

(y)basedonPrandtl’smixing-lengthhypothesisi.e.

σ

(

y

)

=ky. How-ever,duetothegridstretchinginthewall-normal(y),closetothe wall the sizeof theeddy becomestoo smallto bediscretized in spanwise and streamwise direction, therefore the following rela-tionisusedtoestimatethesizeoftheeddies

σ

(

y

)

=max

{

ky,

1 }

(

t

)

+Ucdt (A.6)

wheredtisthetimestepofthesimulation.Whenaneddyleaves the boxB,a neweddy isgenerated onthe inletof thebox with randomspanwiseandwall-normallocations.

Once u˜ is generated on the inlet plane with the appropriate temporal and spatial correlations, one obtains a velocity field u

withtheprescribedmeanandcovarianceprofiles,by linear trans-formationofu˜,(providingthatu˜iu˜j=

δ

i j):

(A.7)

whereUi(y)isthemeanvelocityprofileandAij(y)istheCholesky

decompositionoftheReynoldsstresstensorR_ij(y)profile:

Ai j=





(A.8)

BecausewithSEMtechniconlythelargestturbulentstructures are generated, the flow presents a transient region downstream theinletplanebeforereachingthefullyrealisticturbulent correla-tionsandstructures.Eddiesofsmallersizewerefoundtoundergo

alongtransientandtherefore,requirealongercomputational do-main toreach higherReynoldsnumber.Moreover, awayfromthe wall the smaller structures dissipate rather than to evolve into large-scale structures resulting in unphysical velocity field. How-ever, whenPrandtl’smixing-length hypothesis(seeEq.A.5) isused to provideinhomogeneityinthewall normaldirection,the large-scale structures away fromthe wall breakdown andenergy cas-cadestosmallerscales,therebyprovingmorephysicaleddy distri-butions.

Topreventtheinteractionofbubbleswithunphysicalstructures due tothe transient,we add bubbles afterthetransientand dis-regard the transientlength in furtheranalysis. Tosave computa-tionaltimethesimulationispartitionedintotwoseparate simula-tions ([10]). Theyare referredasDomain-AandDomain-B sim-ulations. The advantage ofthis setup isthat we have toperform

Domain-A simulationonly onceto makea databaseofmore re-alistic inletboundaryconditions.Domain-B,whenprovided with theinletconditionissuedfromtheDomain-A,simulationdoesnot need anytransitionlengthto becometurbulent. Thestrategy im-plementedcouldbesummarizedinthefollowingsteps:

1.Domain-AutilizesSEMtogenerateboundaryconditionatthe inletplane

6

1.

2. Ateverytimesteps ofDomain-Arun,thevelocityfield inthe

y−z plane

6

2 locatedatstreamwise location 12

δ

is savedto

thedisc.

3. Domain-Breadsthisvelocityfieldevery timestepasaninlet boundarycondition.

AppendixB. Verificationandvalidation

We presentthe differenttest casesusedtoverifyandvalidate the newimplementations inourJADIM code.We firstreport the validationoftheturbulentboundarylayersimulationinB.1.Then inB.2themomentumforcing termFd→c hasbeentestedwithout

anyothervoidfractioneffectbysetting

ε

c=1(case1),and even-tually inB.3thedifferentcontributionsofthevariation

ε

c inboth

the continuityandthemomentum equationsareverifiedwithout consideringtheeffectofF_d_→_c (cases2to6).

B1. Validationoftheturbulentboundarylayersimulation

We report herethe validationoftheturbulent boundarylayer simulation.Asexplained,inSectionAppendixA,thesynthetic ed-dies generatedneedsome streamwisedistancetoevolveand pro-vide completelyphysical turbulent structures. Therefore, to avoid

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Fig. B1. (a) Mean flow profiles U +₍_y+₎_{; (b) profile of the variance of the three velocity components u}+

rms , v+rms and w +rms ; (c) Reynolds stress −u ′v′

+

profile. Comparison with the DNS of [23] .

Table B1

Relative error for the pressure gradient as a function of the mean void fraction.

α 5 . 1 × 10 −4 _{5 . 1 × 10}−3 _{5 . 1 × 10}−2

Relative error for 1P 5 . 4 × 10 −6 _{7 . 3 × 10}−6 _{6 . 3 × 10}−6

the interaction of bubbles with these un-physical structures and savecomputationaltime,we havebrokenthesimulationintotwo parts Domain-A - Domain-B (Fig. A2). In Domain-A we per-formed a single phase simulation in order to generate an inlet boundary condition database that will be subsequently used for the two-phase flow simulations carried out in the Domain-B. Table 3 presents the dimension and the mesh resolution of the

Domain-B.Notethat Domain-Aisthree timeslongerin stream-wise direction than Domain-Bandof thesamesize inspanwise andwall-normaldirections.

FigureB1showsthestatisticsofturbulenceforthesinglephase flowcomparedagainsttheDNSresultsof[23]atRe_θ≈1100where

Re_θ=U0

δ

θ/

ν

.Thesestatisticsarecomputedbytemporaland

span-wiseaverageatagivenstreamwiselocation.Thestreamwisemean velocity profileisingoodagreementwith[23].Similarly,the pro-files of the variance of the streamwise and wall-normal velocity componentsareingoodagreement,but,forthespanwisevelocity componentweobserveaslightdiscrepancywhichislikelydueto thesmallerspanwisewidthconsideredinoursimulations.

B2. Verificationoftheimplementationofthemomentumforcing termF_d

→c

We first verifythe momentum forcing term Fd→c by

consid-ering the pressure drop generated by a random distribution of fixed identical bubbles. We consider a 3D channel as shown in the figureB2.Thedimensionsof thechannel areLx,Ly andLz in

the x, yand z directions,respectively. Theyare chosen such that

Lx=5Lyand Lz=1.175Ly.Nbbubbles ofvolume Vbare randomly

distributedandmaintainedfixedinthecentralpartofthedomain

Fig. B2. Sketch of the geometrical configuration used for the verification in the case 1 with N b = 40 , 0 0 0 uniformly distributed bubbles of radius R b = 2 × 10 −4 L y .

Lx/4≤x_b≤3Lx/4.Inthisregionthemeanvoidfractionisthus

α

= NbVb

LxLyLz/2

Threedifferentvoidfractions,

α

=5.1×10−4_,₅_.₁_×₁₀−3_and₅_.₁_×

10−2_, _are _considered. _We _impose _a _constant _inlet _velocity _U₌

U1ex. Stress-free boundary conditions are imposed on the four

boundaries parallel to the mean flow and inflow-outflow condi-tionsintheaxialdirection. Themeshisregularinthexandz di-rectionswhileitisnon-uniformintheydirection.Thenumbersof cellsineachdirectionareNx=50andNy=Nz=64,respectively.

Inthiscase,wedo notconsidertheeffectofthevoidfraction in the system of equations (9–10). By imposing the momentum sourceterm−Fbexforeachbubble,thepressurejumpbetweenthe

inletpressureP_inandtheoutletpressurePoutisgivenby:

(

Pout−Pin

)

S=−NbFb (B.1) P

(

x

)

−Pin 0.5

ρ

fUin2 =− LxFb

α

V_b

_ρ

_f_U2 in (B.2)

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Table B2

Verification tests and corresponding analytic solutions for non-uniform void fraction distributions.

Case Governing Equations Volume fraction Velocity Pressure

(imposed) (Exact solution) (Exact solution)

∂εcUk ∂xk = 0 2 ρf£εc∂_∂U_ti+ U k∂∂Uxki ¤ = εc(x ) = εin / (1 + αx ) U(x ) = U in(1 + αx ) P(x ) = −12ρf(U 2(x ) − U 2(L x)) −_∂∂_xP i+ µ ∂ ∂xk £∂Ui ∂xk+ ∂Uk ∂xi ¤ ∂εc ∂t + ∂εcUk ∂xk = 0 3 ρf£εc∂∂Uti+ U k∂∂Uxki ¤ = εc(t) = εin(1 − t/T o) U(x ) = U in + t−Txo P(x ) = −ρf 1 2 £ U 2 − U 2₍_{x = L} x) ¤ −_∂∂_xP i+µ ∂ ∂xk £∂Ui ∂xk+ ∂Uk ∂xi ¤ − εin 2To(To−t) £ x 2 − L 2 x ¤ ∂εcUk ∂xk = 0 4 ρf£εc∂∂Uti+ U k∂∂Uxki ¤ = εc(x ) = εin / (1 + αx ) U(x ) = U in(1 + αx ) P(x ) = −12ρfUεinin [(1 + αx ) 2 −∂εcP ∂xi +µ ∂ ∂xk £∂Ui ∂xk + ∂Uk ∂xi ¤ −(1 + αL x)2 ](1 + αx ) ∂εcUk ∂xk = 0 5 ρf£εc∂∂Ut + i εc U k∂∂Uxki ¤ = εc(x ) = εin / (1 + αx ) U(x ) = U in(1 + αx ) P(x ) = −ρfαU in2(x − L x)(1 + αx ) −∂εcP ∂xi +µ ∂ ∂xk £∂Ui ∂xk + ∂Uk ∂xi ¤ ∂εc ∂t + ∂εcUk ∂xk = 0 6 εc U k∂∂Uxki = εc(x ) = εin / (1 + αx ) U(x ) = U in(1 + αx ) P(x ) = −ρfεin U 2 inα(x − L x) /ε(x ) −∂εcP ∂xi + µ ∂ ∂xk£εc ¡∂Ui ∂xk+ ∂Uk ∂xi ¢¤ +2 µαU inε(xε)(−x)εLx Table B3

Errors obtained for the different tests reported in the Table B.6 .

Case Max relative error for U Max relative error for P Max error for V Max error for W

2 4 . 0 × 10 −4 _{4 . 80 × 10}−4 ₁₀−11 ₁₀−11

3 3 . 80 × 10 −4 _{3 . 30 × 10}−3 ₁₀−14 ₁₀−14

4 4 . 5 × 10 −4 _{5 . 7 × 10}−4 ₁₀−11 ₁₀−12

5 1 . 0 × 10 −4 _{4 . 0 × 10}−4 ₁₀−11 ₁₀−11

6 4 . 0 × 10 −4 _{3 . 1 × 10}−4 ₁₀−13 ₁₀−13

Fig. B3. Evolution in the x direction of (a) the pressure averaged in the channel section, and (b) the same pressure scaled by the global void fraction.

Fig. B4. Summary of the mean fluid concentration (left), pressure (middle) and velocity (right) for the different cases in the x direction (for case 3 it corresponds to time t = 0 . 05 L y /U in ).

whereS=LyLzisthechannelsection.Infactthepressuregradient

is constant through the bubble cloud and is proportional to the voidfraction