Optimal streaks in the wake of a blunt-based axisymmetric bluff body and their influence on vortex shedding

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Optimal streaks in the wake of a blunt-based

axisymmetric bluff body and their influence on vortex

shedding

Mathieu Marant, Carlo Cossu, Grégory Pujals

To cite this version:

Mathieu Marant, Carlo Cossu, Grégory Pujals. Optimal streaks in the wake of a blunt-based

ax-isymmetric bluff body and their influence on vortex shedding. Comptes Rendus Mécanique, Elsevier

Masson, 2017, 345 (6), pp.378-385. �10.1016/j.crme.2017.05.010�. �hal-02042706�

Optimal

streaks

in

the

wake

of

a

blunt-based

axisymmetric

bluff

body

and

their

influence

on

vortex

shedding

Mathieu

Marant

a

,

Carlo

Cossu

a

,

Grégory

Pujals

b

a_{InstitutdemécaniquedesfluidesdeToulouse,CNRS–INP–UPS,2,alléeduProfesseur-Camille-Soula,31400Toulouse,France} b_{PSAPeugeotCitroën,CentretechniquedeVélizy,2,routedeGisy,78943Vélizy-Villacoublaycedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received3February2017 Accepted11May2017 Availableonline7June2017

Keywords:

Fluiddynamics Hydrodynamicstability Flowcontrol

Wecomputetheoptimalperturbationsofazimuthalwavenumberm THATmaximizethe spatial energy growth in the wake of a blunt-based axisymmetric bluff body. Optimal perturbationswith m=0 leadto the amplification ofstreamwise streaks inthe wake. Whenforcedwithfiniteamplitudem_=1,optimalperturbationshaveastabilizingeffect onlarge-scaleunsteadyvortexsheddinginthewake.Weshowthatm≥2 modes,which are forced withzero mass flux, cansignificantly reducethe amplitude ofthe unsteady lift force exerted on the body. When combined with low levels of base bleed, these perturbationscancompletelysuppresstheunsteadinessinthewakewithreducedlevels ofmassinjectionintheflow.

©2017Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Suitable three-dimensional perturbations applied to nominally two-dimensional basic flows are known to be able to weakenandevensuppressvortexsheddinginthewakeofbluffbodies(see,e.g.,[1,2]and[3]forareview).Ithasrecently beenshownthatthisstabilizingactionisassociatedwiththequenchingofthelocalabsoluteinstabilityinthewake[4,5], leadingtothestabilizationoftheassociatedglobalmode[6,7].

In the caseof 3D control of 2D wakes, the role ofthe stabilizing perturbations is essentially to force spanwise peri-odic perturbationsofthe streamwisevelocityinthewake.Inthe literaturepertainingto wall-boundedshearflows,these spanwise periodic perturbations ofthestreamwise velocity are knownas‘streamwisestreaks’ andthey areknown to be very efficientlyforcedby streamwisevorticesthroughthelift-upeffect(see,e.g.,[8,9]forareview).Theshapeofthe op-timal forcing leadingto themaximally amplified streakscan becomputed throughstandard optimizationtechniquesand is associated withlarge energyamplifications ofthe forcing, whosemaximal value typically increaseswiththe Reynolds number.Suchanoptimizationhasbeenrecentlyperformedonparallelandnon-parallelmodelwakes[5,6] andonthe cir-cularcylinderwake[7],showingthat theefficiencyofthe3D controlof2Dwakescanbegreatlyimprovedbyforcing the streaks optimally.In thecaseof thecylinder2D wake, itis foundthat the optimalspanwisewavelengths leading tothe mostamplifiedstreaksinthewakealmostcoincidewiththeonesthatarethemostefficienttoquenchvortexshedding[7]. Whileimportantprogresshasbeenachievedinthecaseofthe3Dcontrolofnominally2Dwakes,thesameisnottrue in thecaseofthree-dimensional wakes,whicharethe mostrelevantto manyapplications(see,e.g.,[10] fora discussion of this issue inthe context ofthe aerodynamics of heavy vehicles). The scope ofthe present studyis therefore to test

E-mailaddress:carlo.cossu@imft.fr(C. Cossu).

http://dx.doi.org/10.1016/j.crme.2017.05.010

1631-0721/©2017Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Fig. 1. Longitudinalsectionoftheaxisymmetricblunt-basedcylinderofdiameterD withaxisx paralleltothefree-streamvelocity.Thebluntnoseofthe cylinderisahalf-ellipsoidofcircularcross-section(diameterD)andlongitudinalhalf-axisoflengthD extendingfromx/D= −1 tox/D=0.Thetubular bodyhasadiameterD andextendsfromx/D=0 tox/D=1.

theeffectivenessofan extensionto 3Dwakes oftheapproachusedfor2D wakes.Wechoose asa testbedthewakeofa blunt-basedaxisymmetricbluffbodywithanellipsoidalnoseandasquarebackwhoseglobalstabilityhasbeenpreviously investigated[11].Forthisconfiguration,ithasbeenfoundthatthesequenceofglobalinstabilitiesdevelopinginthewakeis similartotheoneobservedforasphereandinotheraxisymmetricwakeswithafirststeadyinstabilityoftheaxisymmetric wake that breaks the axisymmetry by giving rise to a non-zero steady lift force. This primary state then undergoes a (secondary) instability leading to unsteadiness in the wake and to an unsteady liftforce on the body. It has also been shownthatfortheconsideredconfigurationtheseglobalinstabilitiescanbestabilizedwithbasebleed[11].Ourapproach, similarly to previous investigations [5–7,12–15], consistsin first computing the steady, azimuthally (‘spanwise’) periodic optimal perturbations inducing the maximum growth of streaks in the wake and then study their stabilizing effect on the wakeunsteadiness. We anticipatethat thefound m

=

1 optimal perturbations havea stabilizingeffecton the global instabilities.

Themathematicalformulationoftheproblemisintroducedin§2.Thecomputedoptimalenergyamplificationsandthe associated perturbations aswell asthe analysisof their stabilizing effect are presented in §3. These results are further discussedin§4wheresomeconclusionsarealsodrawn.

2. Problemformulation

We considertheflow ofan incompressibleviscous fluidofdensity

ρ

andkinematicviscosity

ν

pastan axisymmetric blunt-basedcylinderofdiameterD andtotallengthL

=

2D whoseaxisisparalleltothefree-streamvelocityU_∞ex (where

ex is theunit vector orientedparallel to the axis x ofthe cylinder). The blunt nose ofthe cylinder is an ellipsoid with

circularcross-section of diameter D andlongitudinal half-axiswith2:1 ratio.The tubular body has adiameter D anda length D (see Fig. 1). Indimensionlesscoordinates based on D, therefore,thenose ofthe bodyextends fromx

= −

1 to x

=

0,thetubularbodyfromx

=

0 tox

=

1,andthewakeoccupiestheregionx

>

1.

TheflowisgovernedbytheNavier–Stokesequationsforanincompressibleviscousfluid:

∇ ·

u

=

0

,

∂

u

∂

t

+

u

· ∇

u

= −∇

p

+

1 Re

∇

2_{u (1)}

whereu and p andthedimensionlessvelocityandpressurefieldsand Re

=

U_∞D

/

ν

istheReynoldsnumber.Thevelocity, pressure, lengths and times have been made dimensionless with U_∞,

ρ

U_∞2 , D and D

/

U_∞ respectively. Homogeneous boundary conditions forthe velocity are enforced on the body surface exceptin the controlled case wherewall-normal controlvelocitiesareenforced.

Inthefirstpartofthestudy,wecomputethelinearoptimalspatialperturbationsofthesteadyaxisymmetricsolutionto theNavier–StokesequationsU0,whichislinearlystableforsufficientlylowReynoldsnumbers.Theseperturbations satisfy

theNavier–Stokesequationsrewritteninperturbationform:

∇ ·

u

=

0

,

∂

u 

∂

t

+

u 

_{· ∇}

_U

₊

_U

_{· ∇}

_u

₊

_u

_{· ∇}

_u

_{= −∇}

_p

₊

1 Re

∇

2_u ₍₂₎

whereU

=

U0 andthenonlineartermu

· ∇

uisneglectedforthecomputation ofthelinearoptimalperturbations.Inthe

following, steadyperturbations u are considered, which are ofparticular interest inopen-loop flowcontrol applications andwhichareforcedbyradialblowingorsuctionu_w

(θ,

x

)

erenforcedonthebodylateralsurface(0

<

x

/

D

<

1,r

/

D

=

1

/

2).

Similarlyto[7],theoptimalspatial energyamplificationofwallcontrolforcingisdefinedasG

(

x

)

=

maxuwe

(

x

)/

ew,where

ew isthe (input)kinetic energyoftheblowingandsuction forcedonthe lateralsurface ande isthe(output) local

per-turbationkineticenergyatthestationx respectivelydefined,indimensionlesscoordinates,ase_w

= (

1

/

4

)



₀2π



₀1

(

u_w

)

2dxd

θ

Fig. 2. Longitudinal section of the grid showing the increased grid density in the regions of higher shear.

To numerically compute G

(

x

)

and the associated optimal wall perturbation, we follow an approach similar to the one recently applied to the circular cylinder wake [7]. The (control) radial velocity, enforced on the lateral cylindrical surface uw

(θ,

x

)

is decomposed on a set of linearly independent functions b

(n)

w, in practice truncated to N terms, as:

u_w

(θ,

x

)

=



Nn=1qnb(wn)

(θ,

x

)

. Denoting by b(n)

(θ,

r

,

x

)

the perturbation velocity field obtained by using b(wn)

(θ,

x

)

as

in-put,fromlinearity it followsthat u

(θ,

r

,

x

)

=



nN=1qnb(n)

(θ,

r

,

x

)

.The optimalenergygrowthcanthereforebe computed

withitssubspaceapproximationG

(

x

)

=

maxqqTH

(

x

)

q

/

qTHwq,whereq istheN-dimensionalcontrolvectorofcomponents

qn and the components of the symmetric matrices H

(

x

)

and Hw are defined as Hnj

(

x

)

= (

1

/

2

)



_∞ 0



2π 0 b(n)

·

b(j)r d

θ

dr; Hw,mn

= (

1

/

4

)



2π 0



1 0b (m)

w

(θ,

x

)

b(wn)

(θ,

x

)

d

θ

dx. The optimal energy growth G

(

x

)

is easily found asthe largest eigenvalue

μ

maxofthegeneralizedN

×

N eigenvalueproblem

μ

Hww

=

Hw.Thecorrespondingeigenvectorq(opt)isthesetofoptimal

coefficientsmaximizingthekineticenergyamplificationattheselectedstreamwisestationx,andthecorrespondingoptimal blowingandsuctionisgivenby u (wopt)

(θ,

x

)

=



N n=1q

(opt)

n b(wn)

(θ,

x

)

.Inthefollowing,asthebasicflowU0 isaxisymmetric

and theequations linear,our results areobtained by computingindependently thesingle-harmonic azimuthally periodic perturbations u_w

(θ,

x

,

m

)

=

f

(

x

)

cos m

θ

. The maximum growth is finally defined as Gmax

=

maxxG

(

x

)

, andis separately

computedforeachconsideredazimuthalwavenumberandconvergedbyincreasingthetruncationorder N.

In the second part of the study, the effect of three-dimensional optimal perturbations on the wake unsteadi-ness is investigated by forcing them with finite amplitude Aw, and therefore enforcing the following radial

veloc-ity at the wall in the nonlinear numerical simulations: uw

(

x

,

m

)

=

Awf(opt)

(

x

,

m

)

cos m

θ

, where f(opt) is normalized

so as to obtain e_w

= (π/

2

)

A2

w. The cost of the control can be quantified with the momentum coefficient of

forc-ing Cμ

=



Slatu 2 wdS

+



Sbaseu 2 bdS



/(

π

R2U2_∞

/

2

)

, which, using dimensionless units, is Cμ

=

8 A2w

+

2Cb2. Even more

im-portantly in applications, the cost of the control can also be quantified in terms of dimensionless mass flux CQ

=



SlatuwdS

+



SbaseubdS



/(

π

R2_U

∞

)

. Whenoptimal perturbations withm

>

0 are used,the associated massflux is zero and therefore CQ

=

Cb. However, for m

=

0, the mass flux is non-zero and in this case, in dimensionless units CQ

=

Cb

+

4 Aw



1 0 f

(opt)

₍

_x

_,

_m

₌

₀

₎

_{dx,wherepositiveandnegative A}

w areassociatedwithblowingandsuction,respectively.

Allthe presentedresultsare basedonnumericalintegrations ofthefull orthelinearizedNavier–Stokesequations per-formedusingacustomizedversionofOpenFoam,anopen-sourcefinitevolumecode(seehttp://www.openfoam.org),which hasalreadybeenvalidatedandusedinaseriesofpreviousinvestigationsinourgroup[5–7].Theflowissolvedina three-dimensionaldomainextending2

.

5D upstreamofthebody,3D fromthesymmetryaxis,and10D downstreamofthebody stern.The gridisstructured andrefined nearthe bodysurface, asshowninFig. 2.The PISOandSIMPLEalgorithms have beenrespectivelyusedtoadvancethesolutionintimeandtocomputesteadysolutions.

3. Results

Thedifferentregimesofthewakeobservedintheuncontrolledcase(nonormalvelocityforcedonthebodysurface)have been thoroughlyinvestigatedin[11].Asteadyaxisymmetricflow isobservedforReynoldsnumbers below Re1



319.At

Re

=

Re1,theaxisymmetryflowbecomesgloballyunstableandisreplacedbyasteadyflowcharacterizedbythepresence

oftwo(steady)counter-rotatingvorticesoriginatedinthenearwakeandthatdiffusefurtherdownstream.Thisnewsteady flowbecomesunstableatRe2



413.PeriodicvortexsheddingisobservedjustabovethissecondcriticalReynoldsnumber.

Ournumericalsimulationsretrievethesedifferentregimes,asshowninFig. 3wherethevorticalstructurescorresponding to Re

=

300, Re

=

350, Re

=

415,and Re

=

500 are shown. The signaturesofthesedifferentnumericalregimes arewell recognisableontheliftforceexertedonthebody,asshowninFig. 4.Whiletheliftiszerointheaxisymmetricregime,it rises toa non-zerosteadyvaluein therange Re1

<

Re

<

Re2 and,for Re

>

Re2,beginstooscillate aroundthat non-zero

Fig. 3. Vorticalstructuresonthe surfaceandthe wakeofthebluffbodyintheuncontrolledcaseindifferentdynamicalregimesvisualisedwiththe

Q=0.001 surfaces.(a) AtRe=300,theflowissteadyandaxisymmetric.(b) AtRe=350,asteady(non-axisymmetric)flowwithtwocounter-rotating streamwisevorticesinthewakeisobserved.(c) AtRe=415,aperiodicglobalmodeoscillatesontopofthecounter-rotatingvortices,while(d) atRe=500 additionalstructuresofsmallerscalesandhigherfrequenciesenterthepicture.

Fig. 4. TemporalhistoryoftheliftcoefficientassociatedwiththefourdynamicalregimesreportedinFig. 3.Thepermanentregimeisobservedfort>≈250. AtRe=300 theflowissteadyandaxisymmetricandthereforehaszerolift.AtRe=350,thesteadynon-axisymmetricsolutiondisplaysasteadyliftwhich atRe=415 oscillatesperiodically.AhigherleveloftheoscillationsandadditionalfrequenciesareobservedatRe=500.

meanvalue. Theamplitude oftheoscillationsandtheir frequencycontentincrease whentheReynoldsnumberisfurther increased.

Having summarized the ‘reference’ uncontrolled dynamics of the wake, we next consider the optimal energy ampli-fications that can be supported by the axisymmetric wake steady solution U0. In the axisymmetric case, the optimal

amplificationsandtheassociatedoptimalperturbations canbe separatelycomputedforthedifferentazimuthal wavenum-bers.Wethereforecomputetheoptimalspatialamplificationofthesteadyradialvelocity forcingu_w

(θ,

x

,

m

)

=

f

(

x

)

cos m

θ

ofazimuthalwavenumberm appliedonthebodylateralskinatRe

=

300

<

Re1.Followingtheproceduredescribed in§2,

thestreamwisedistribution f

(

x

)

for0

<

x

<

1 isapproximatedwithanexpansiononChebyshevpolynomialsTn

(ξ )

(where

ξ

=

2x

−

1)truncatedtoN terms.Similarlytowhatwas foundinpreviousstudies[6,7],theoptimalamplificationisfound toconvergetoa1%precision withN

∼

O

(

10

)

terms,asshowninFig. 5.Them

=

1 modeisfoundtobethemost ampli-fied (withGmax

>

500). However, asthismode becomes linearlyunstable(with a zerofrequency) abovethe firstcritical

Reynoldsnumber Re1 andthereforehasadestabilizing actionathigherReynoldsnumbers,wedonot furtherconsiderit

inthefollowing.ThetwomostamplifiedmodesatRe

=

300 (exceptedthem

=

1)arethem

=

2 andthem

=

0,whilethe m

=

3 and m

=

4 are muchlessamplified,asshowninFig. 6a.Asexpected, theamplificationsincreasewiththeReynolds number(asshowninFig. 6bforthem

=

2 mode).

Thedynamicsofthem

=

0 (axisymmetric)modeisqualitativelydifferentfromthatoftheothermodesbecause,contrary to them: (a) it is not related to the amplification of streaks (the lift-up effect) but to the Orr mechanism [8], (b) it is associated witha non-zero mass flux, and(c)opposite effects are obtainedwith negative orpositive amplitudesof uw.

Thisdifferentbehaviour isalsorecognisableonthelongitudinalshapes

|

f(opt)

_|(

_x

₎

_{oftheoptimalblowingandsuctionthat}

Fig. 5. (a)ConvergenceoftheG(x)optimalspatialenergygrowthcurvewiththenumberN oflinearlyindependentdistributionsofwallblowingand suctionincludedintheoptimizationbasisforthem=2 perturbationsat Re=300.(b) CorrespondingconvergenceofthemaximumamplificationGmax

withN.ConvergedresultsareobtainedwithN≈6 terms.

Fig. 6. DependenceoftheoptimalgrowthcurveG(x)ontheazimuthalwavenumberm atRe=300 (panela)andontheReynoldsnumberforthem=2 mode(panelb).

Fig. 7. Longitudinalshape|f(opt)_{|(x)oftheoptimalblowingandsuctionat Re=300,whereallthefunctionshavebeennormalizedbytheirmaximum}

absolutevalue.

spanwise periodic blowing andsuction increasingly concentrated near the trailing edge, while for the m

=

0 mode the optimaldistribution

|

f(opt)

_|(

_x

₎

_{isnon-negligibleintheupstreamhalfofthelateralsection.Thevariationsof}

_|

_f(opt)

_|(

_x

₎

_with

Re aresmall(notshown),similarlytowhat wasfound inthe2D caseonthe circularcylinder[7].Thespanwiseperiodic optimal blowingandsuctionofm

=

0 modesinduces counter-rotating streamwisevortices that decaydownstream while forcingthegrowthofstreamwisestreaks,asshowninFig. 8.

Havingcomputedtheoptimalperturbationsleading totheoptimalenergygrowthinthewake,we nextconsidertheir effect onthe unsteady wakefor Re

>

Re2. Tothisend, for eachconsidered azimuthal mode m

=

1, we selectthe shape

ofthe optimalforcing leading tothemaximumamplification Gmax andwedoforce itwithfiniteamplitude Aw.Wefind

that thisforcinghas(form

=

1) astabilizingeffectontheunsteadiness ofthe wakeasmeasured byvariations ofthelift coefficient CL.Forthem

=

0 forcingthe stabilizingeffectisobtainedwithsuction( Aw

<

0), whileblowing( Aw

>

0) has

theopposite effect(notshown).Theeffectoftheforcing onthe CL

(

t

)

,showninFig. 9forthem

=

2 mode,istoleadtoa

permanentreductionofboththeamplitudeoftheliftoscillationsandtheirmeanvaluewhentheamplitude Awofoptimal

blowing andsuction is increased. The stabilizing effect offorcing optimal perturbations is associated withan increasing ‘symmetrization’ofthewakeinducedbytheforcedstreaks,asshowninFig. 10.

Fig. 8. Cross-stream( y−z)viewofthevelocityperturbationsforcedbythem=2 optimalblowingandsuctionatRe=300 atthethreeselectedstreamwise stations:x=1 (bluff-bodystern,panela),x= (xmax+1)/2 (midwaytothepositionofmaximumstreakamplitude,panelb)andx=xmax (positionof

maximumstreakamplitude,panelc).Thescalesusedtoplotthecross-streamv–wvelocitycomponents(streamwisevortices,arrows)andthestreamwise

ucomponent(streamwisestreaks,contourlines)arethesameinallpanels.Thecircularcrosssectionofthebaseofthebluffbodyisalsoreportedasa (red)circleforreference.

Fig. 9. TemporalhistoryoftheliftcoefficientassociatedwiththeuncontrolledflowatRe=500 ( Aw=0,asalsoreportedinFig. 4)andwiththeincreasing

amplitudesAwofthem=2 optimalblowingandsuctionatRe=500.

Fig. 10. Vorticalstructures(visualisedwiththe Q=0.001 surfaces)onthesurfaceandthewakeofthebluffbodyatRe=500 intheuncontrolledcase (panela,whichisthesameaspaneld ofFig. 3)andwiththem=2 optimalblowingandsuctionenforcedwithAw=0.014 (panelb)and Aw=0.028

(panelc).

Similarresultsareobtainedfortheother(m

=

1) modes,asshowninFig. 11,wherethedependenceoftherootmean square amplitude oftheliftcoefficient oscillationsC_L(rms) onthecontrol amplitude measuredin termsofthemomentum coefficient Cμ is reported. Complete stabilization can be obtainedat Re

=

415 (near the critical Reynolds number) and significant reductionsofthe‘rms’amplitudes canbe achievedat Re

=

500.However, excessiveamplitudesof thecontrol forcingcanresultinanewincreaseinC_L(rms),nowsustainedbytheinducedstreaks.

Asalreadymentioned,fortheconsideredflow, standardbasebleedisaneffectivewaytosuppressunsteadinessinthe wake[11].We thereforecomparetheeffectobtainedwiththe selectedazimuthal modesofoptimalblowingandsuction at Re

=

500 tothe purebase bleedinFig. 12a. From thisfigure,itisseen how, amongtheoptimalblowingandsuction controls, the least effective mode is the axisymmetric one m

=

0. The m

=

2 mode is the mosteffective atlow forcing amplitudes,whilem

=

3 ismoreeffectiveathigherforcingamplitudes.

Fig. 11. DependenceoftheC(Lrms)rootmeansquareamplitudeoftheliftcoefficientoscillationonthecontrolamplitudemeasuredintermsofthe

momen-tumcoefficientCμforRe=415 (diamonds)and Re=500 (triangles)forthem=0 (panela),m=2 (panelb)andm=3 (panelc)optimalblowingand

suction.

Fig. 12. ComparisonoftheCL(rms)(Cμ)dependenceofthem=0,m=2 andm=3 optimalblowingandsuctionintheabsenceofbasebleed(panela)and

withbasebleedCb=0.01 (panelb)andCb=0.02 (panelc).Forconvenience,thesecurvesarecompared,inallpanels,totheoneobtainedbypurebase

bleed(BB)intheabsenceofanyforcingofoptimalblowingandsuction( Aw=0).

Purebasebleed( Aw

=

0)isfoundtobemoreeffectivethantheoptimalblowingandsuction,butatthecostofasteady

mass injectioninthe flow(while m

=

0 optimalblowingandsuction are associatedwithzeromass flux). Itis therefore interesting totest ifcombinationsofbasebleedandoptimalblowingandsuctioncould leadtothecompletesuppression oftheunsteadinessusinglowerlevelsofmassinjectioninthewake.Thisis,actually,alsointeresting forthem

=

0 mode, which,beingassociatedwithsuction,couldprovideatleastpartofthemassusedforbasebleed.Wehavethereforeexplored ifcombinationsofstandardbasebleedandoptimalblowingandsuctioncould enhancethecontrolperformance.The first twobasebleedvaluesreportedinFig. 12a(Cb

=

0

.

01 andCb

=

0

.

02,correspondingtothesecondandthirdpointfromthe

rightontheblack-diamondcurve)havethereforebeenusedincombinationwiththeoptimalblowingandsuctionforcing, asreportedinpanelsbandcofFig. 12.Fromthesefigures,itisseenhowthecombinationofoptimalblowingandsuction andbasebleed isefficient,leadingto thecompletestabilizationoftheoscillations inthewakewithamassflux reduced whencomparedtothepurebasebleedwiththesameCμ.

4. Conclusions

Thescopeofthisstudywastwo-fold:(a)compute,forthefirsttime,theoptimalsteadyperturbationsofa3D (axisym-metric)wakeinducedbyoptimalblowingandsuctionontheskinofthebodyand(b)analyseiftheseoptimalperturbations have a stabilizingeffect onthe wake whenforced withfiniteamplitude. In its scope andmethods, thepresentstudy is thereforeanextensionto3Dwakesoftheapproachsuccessfullyimplementedon2Dwakes[6,7].

It is found that the steady (m

=

0) optimal blowing and suction leads to the formation of streamwise vortices that induce the growth ofstreamwise streaks,like in 2D wakes. Them

=

1 mode isthe mostamplified for Re

<

Re1, butit

becomes linearlyunstable for Re

>

Re1. Other modes are alsoamplified but, contrary to what found in2D wakes, they

are associated withmaximum amplifications that are smaller than those found in the 2D case on the circularcylinder. This is not extremelysurprising because, in the 2D circular cylinder wake, the most amplified mode corresponds to a spanwisewavelength

λ

z

≈

5

−

7D (where D isthecylinderdiameter),whileshorterwavelengthsaremuchlessamplified.

As intheaxisymmetriccase,theanalogousofthespanwisewavelengthis

λ

z

≈ π

D

/

m,themaximumaccessible‘spanwise

wavelengths’ are thereforesmall, e.g.,

λ

z

/

D

≈

1

.

5 form

=

2.The maximumamplifications achievablein anaxisymmetric

wake are thereforearguablysmall ifan analogywiththe 2D wakeis made. Anotherconsequenceofthe relatively small accessible

λ

z isthat thestreamwisepositionwhere themaximumgrowthisattainedisalsosmaller thaninthe2D case

at comparable Reynolds numbers. Similarly to 2D wakes, the longitudinal shape of the optimal blowing and suction is not sensitive to the azimuthal wavenumber m (except for them

=

0 mode whose amplificationis based on a different mechanism).

Wehavealsoshownthat,whenforcedwithfiniteamplitude,m

=

1 optimalblowingandsuctionhasastabilizingeffect on theunsteadiness inthe wake,reducing themeanandthe fluctuatingamplitude ofthe lift.When combinedwiththe usualbasebleedonthebodybase,thistypeofcontrolcanleadtothecompletestabilizationofthewakeevenatRe

=

500. Asthestabilizingoptimalblowingandsuction isassociatedwithazeromassflux form

>

0 andtonetsuction form

=

0, whencombinedwithbasebleed,thestabilizationisobtainedwithsmallermassfluxesthaninthe purebasebleed case. Furtherinvestigation isunderwaytofind ifacombinedoptimisation includingthebasebleed shapecan lead tofurther improvementsofthiscontrolstrategy.

Acknowledgement

FinancialsupportfromPSAPeugeot–Citroëniskindlyacknowledged.

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