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

(1)

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rchive

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To cite this version :

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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(2)

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

The aim of this study is to investigate numerically the interaction between a dispersed phase composed of micro-bubbles and a turbulent boundary layer flow. We use the Euler–Lagrange approach based on Direct Numerical Simulation of the continuous phase flow equations and a Lagrangian tracking for the dispersed phase. The Synthetic Eddy Method (SEM) is used to generate the inlet boundary condition for the simulation of the turbulent boundary layer. Each bubble trajectory is calculated by integrating the force balance equation accounting for buoyancy, drag, added-mass, pressure gradient, and the lift forces. The numerical method accounts for the feedback effect of the dispersed bubbles on the carrying flow. Our approach is based on local volume average of the two-phase Navier–Stokes equations. Local and temporal variations of the bubble concentration and momentum source terms are accounted for in mass and momentum balance equations. To study the mechanisms implied in the modulation of the turbulent wall structures by the dispersed phase, we first consider simulations of the minimal flow unit laden with bubbles. We observe that the bubble effect in both mass and momentum equations plays a leading role in the modification of the flow structures in the near wall layer, which in return generates a significant increase of bubble volume fraction near the wall. Based on these findings, we discussed the influence of bubble injection methods on the modulation of the wall shear stress of a turbulent boundary layer on a flat plate. Even for a relatively small bubble volume fraction injected in the near wall region, we observed a modulation in the flow dynamics as well as a reduction of the skin friction.

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

(3)

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

(

ρ

_f−

ρ

_b

)

g (where

φ

v istheglobal

voidfraction,

ρ

f and

ρ

b thedensityof thefluid andthebubbles

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

(4)

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. SEM givesa 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:

d x d t = v (3)

ρ

bVb d v d t =

(

(4)

whereCDisthedragcoefficient,CM theaddedmasscoefficient,CL

istheliftcoefficientandthehistoryforceisneglected[34].

Bubble shapeisassumedtobesphericalinthisstudy.The

as-sumption hasbeena posteriori validated by calculatingfromthe

simulationsthebubbleWebernumberWe=

ρ

d

|

v−u

|

2 d/

σ

based

ontheslipvelocity

|

v−u

|

.Itsmaximumvaluehasbeenfoundto

beoftheorderof10−2_._Hence,_the_drag_coefficient_C

Discalculated

throughthecorrelationof[39]

CD= 16 Reb

"

1 +

µ

8 Reb +1 2

µ

1 + 3 . 315 Re1/2 b

¶¶

−1

#

(5)

wherethebubbleReynoldsnumberisdefinedas

Reb=

|

u−v

|

2 Rb

ν

, (6)

theadded-masscoefficientisCM=1/2andtheliftcoefficientCL is

givenbytherelationof[31]:

CL=

·

1 . 8802

(

1 + 0 . 2 Reb/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

(5)

each bubblelocation.Weusedsecond-orderaccuracylinear

inter-polationscheme.Athird-orderRunge-Kuttaschemeisusedfor

in-tegratingtheforce balanceintime withatypical timestepequal

to one fifth of the viscous relaxation time of the bubbles. Loop

nesting is used when time steps of both phasesare widely

sep-arated. When the Eulerian time step (based on numerical

stabil-itycriteria)issmallerthantheLagrangiancharacteristictimethen

solutions ofbothsetsofequationsaresynchronized.Forlarge

ra-tioofthesetimesteps

1

tEuler/

1

tLagrange,Lagrangianinnerloopsare

integratedinafrozenflowfieldandtheglobalcomputingtimeis

dominatedbytheLagrangiansolverforO

(

105₋₁₀6

₎

_bubbles.

Typ-ically,wehave

1

tEuler/

1

tLagrangeC20.

TheLagrangiansolverJADIMhasalsobeenvalidatedandused

in thepast forthe simulationof bubblesinduced convection [5],

bubblemigrationinapipeflow[30]andbubblepreferential

accu-mulationinCouette-Taylorflow[3,6].

2.3. Thetwo-waycouplingmodel

To account forthe feedback effect ofthe dispersed phase on

the carrying fluid flow, we average the Navier-Stokes equations

in acontrol volume offluidpopulated bymanybubbles. The

ba-sic principles of this operation can be found in several previous

works such as [1,7,9]. The averaging procedure is based on the

function characterizing the presence of the continuous phase

χ

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

)

d 3

ξ

=

h

χ

c

i

V, (8)

whereVisthevolumeofthecells.Wenotethatthecorresponding

void fractionis computed as

ε

d

(

x,t

)

=1−

ε

c

(

x,t

)

. In our study,

the voidfractionassociatedwiththecontrol volumeis estimated

byaddingthevolumeofeachbubblecontainedinthiscontrol

∇

Uc

)

T

¤

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

· ρ

bVb

(

d vb d t −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

ε

c=1ineq9andeq10

(6)

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

ineq9andeq10

mfu2:Onlythevoideffectisconsideredincontinuityequation (9)via

ε

cwithFd→c=0ineq10and

ε

c=1ineq10

mfu3:Onlythevoideffect(

ε

c6=1)isconsideredinmomentum

equation(10)withFd→c=0ineq10and

ε

c=1ineq9

mfu4: The void effect (

ε

c6=1) is considered both in

continu-ityandmomentumequationsandthefeedback(F_d_→_c)term

fromthedispersedtocontinuousisalsoaccountedfor.

Thedescriptionofthedomainsizeandparametersusedforthe

two-phase simulation is given in Table 1 and are similar to the

configurationproposedinJiménezandMoin[24].TheStokes

num-berisdefinedasSt=

τ

b/

τ∗

,where

τ

bisthebubblerelaxationtime

defined as

τ

b=R2b/6

ν

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 caseofthe mfu4 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 therefore mfu0 could 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.Weremarkthattheprofileobtainedfor mfu0 and

mfu1 are very close and decrease towards the wall whereas in the mfu 2, 3, 4 thereisasharpincreaseinthebubble

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

thebubbleasafunctionofwallnormaldistanceforthe mfu0 and

mfu4 . It appears that in mfu4 ,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

casesexcept mfu1 turbulenceintensityishigherthansinglephase

case and the peak is slightly shifted towards the wall. Similarly,

Reynoldsstressprofileisalsomodifiedduetothepresenceofthe

bubble, higherturbulent stress intensityis observed in mfu2,3,4

ascomparedto mfu0 and mfu1 (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

isreducedin mfu4 ascomparedto mfu0 .

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 mfu2 and mfu4 asreported 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

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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 case Random 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

δ0

, for which Re_θ≈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-tionsareslightlylowerthanthe Single-phase case,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 and Single-phase case. 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

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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.

4.4. Quadrantanalysis:Burstingandsweepingevents

Intensity andfrequency ofbursting andsweep eventsare the

maincontributortotheReynoldsstress.Inordertounderstandthe

modificationofReynoldsstress,theinfluenceofmicro-bubbleson

the bursting cycleis investigated using quadrant analysis.

Quad-rant analysis computes the fractional contribution of the

turbu-lent fluctuations to the Reynolds stress at each point inthe

do-main.Theu−

v

planeisdividedintofourquadrants;Q1represents

u >0,

v

_>₀_,_where_high-speed_fluid_rushes_away_from_the_wall,_Q2

containsu<0,

v

_>0_,wherelow-speedfluidmovesawayfromthe

wall, this is generally calledbursting, Q3 represents u<0,

v

_<0

eventswherelowspeedfluidmovestowardsthewallandQ4

con-tains u>0,

v

_<0 events where high-speed fluid rushes towards

thewall, usually referred assweep events.Therefore, Q2 andQ4

contributeto thenegativepartofReynoldsstress andQ1andQ3

bringapositivecontribution.Contribution fromQuadrant2and4

isgenerallymuchhigherthanQuadrant1and3.

InFig.9weshowthefractionalcontributionofthesefour

quad-rants as a function of wall-normal coordinateat the streamwise

location of x=6

δ0

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

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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

λ2

criteria ([22]). It

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

S2+

Ä

2whereSand

_Ä

arethesymmetricandantisymmetricparts

<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

b

λ2(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

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At thedistanceto thewall corresponding tothe bubbleinjection

there isa sudden variationinPb

λ2 butfurtheraway,sayy

+_>₁₀_,

bubbles accumulate in

λ2

<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)

TheSyntheticEddyMethod SEM techniqueisbasedonthe

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)

where

ǫ

k _accounts_for_the_positive_or_negative_contribution_to_the

z

´

(A.2)

fhasacompactsupporton[−1,1]andsatisfiesthenormalization

condition

Z1

−1

f2

(

x

)

d x= 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

₃

2

(

1 −

|

x

|

)

, x ≤1

0 , otherwise (A.4)

Here

σ

x,

σ

y,

σ

zcontrolthesizeoftheeddygenerated.Itis

ad-visedtochoosethelengthscalesclosetotheonefoundinreal

tur-bulentflowasitstronglyaffectstheturbulentfluctuations

gener-ateddownstreamoftheinlet(see[43]).However,differentchoices

forlengthscaleshavebeeninvestigatedin[19]and[21]and

sensi-tive dependence ofthe results onthem is discussed indetail. In

the present study we have chosen length scales to be isotropic

(

σ

x=

σ

y=

σ

z=

σ

(

y

)

) 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 }

(A.5)

constantvelocityUccharacteristicoftheflowusingTaylor’sfrozen

turbulencehypothesis.Ateachtimestepthenewpositionsofthe

eddiesaregivenby

xk

(

t + dt

)

= xk

(

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)

Becausewith SEM techniconlythelargestturbulentstructures

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 referredas Domain-A and Domain-B

sim-ulations. The advantage ofthis setup isthat we have toperform

Domain-A simulationonly onceto makea databaseofmore

re-alistic inletboundaryconditions. Domain-B ,whenprovided with

theinletconditionissuedfromthe Domain-A ,simulationdoesnot

need anytransitionlengthto becometurbulent. Thestrategy

im-plementedcouldbesummarizedinthefollowingsteps:

1.Domain-A utilizes SEM togenerateboundaryconditionatthe inletplane

6

1.

2. Ateverytimesteps of Domain-A run,thevelocityfield inthe

y−z plane

6

2 locatedatstreamwise location 12

δ

is savedto

thedisc.

3. Domain-B readsthisvelocityfieldevery 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

(13)

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-A isthree timeslongerin

stream-wise direction than Domain-B andof 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