Skin friction drag reduction over staggered three dimensional cavities

an author's

https://oatao.univ-toulouse.fr/21303

https://doi.org/10.1016/j.ast.2018.11.001

Gowree, Erwin Ricky and Jagadeesh, Chetan and Atkin, Christopher J. Skin friction drag reduction over staggered

three dimensional cavities. (2019) Aerospace Science and Technology, 84. 520-529. ISSN 1270-9638

Skin

friction

drag

reduction

over

staggered

three

dimensional

cavities

Erwin

R. Gowree

a

,

Chetan Jagadeesh

b

,

Christopher

J. Atkin

b a_{Dept.ofAerodynamics,EnergeticsandPropulsion,ISAE-SUPAERO,UniversitédeToulouse,France} b_{Dept.ofMechanicalEngineeringandAeronautics,CITY,UniversityofLondon,UK}

a

b

s

t

r

a

c

t

The effectof three-dimensionalstaggeredcircular cavities onazero-pressure gradient incompressible turbulentboundarylayerwasstudied.Twokeyparameterswerevaried,beingtheratioofthediameter, d,tothedepth,h,ofthecavity,d/h andtheReynoldsnumberbasedonthediameterofthecavity, Rd.

Velocity profile measurements showed thatfor the casesofd/h>1 anincreaseinskinfriction drag was experienced with respect to asmooth surface, but ford/h≤1 the drag increment was almost negligibleandinsomecasesit waslowerthanthatofasmoothsurfacebyupto10%.Measurements alongthespanwiseplaneshowedthepresenceoforganisedtransversevelocitycomponentswhichbear someresemblancewiththeflowoverriblets.Theskinfrictiondragappearstobeastrongfunctionof Rd,whereforRd>5500 adragincrementisexperiencedwhichcouldpotentiallybeduetoshearlayer

breakdownandmoreproductionofturbulence.

1. Introduction

From the pioneering studies ofNikuradse [15] surface rough-nessorotherexcrescencessuchasgroovesandcavitiesaremajor sources ofskinfriction drag inturbulentboundary layers inwall boundedflows. Nikuradsealsoderived asetofsemi-empirical re-lations to assist in estimating the drag increment with respect to ahydrodynamically smooth surface. Since then a vastamount of research has been conducted which supports the findings of Nikuradseandalsoprovidesfurtherinsightintothephysical mech-anisms, improving the initial semi-empirical and later numerical models.Acomprehensivereviewofturbulentflowoverroughness waspresentedbyJiménez[11].Anewperspectivewasintroduced by Walsh [19] in the 1980s who demonstrated that grooves, or ‘riblets’ alignedwiththestreamwise direction,ifsufficiently sub-merged in the viscous sub-layer, could reduce the viscous drag by almost 10%. Additional experimental validation was provided by Choi [5] in the late 1980s including details of the physical mechanismbehind ribletcontrolandfurtherstudies followed,by Berchertandco-workers[2],[3] and[4],overaperiodamorethan two decades using highly viscous fluid to assist in resolving the flowstructures.

In a reversal of bio-inspired thinking it was thought that, if the riblet mechanism was effective then nature would have got

there first andthistriggered theidea that the mechanismmight already be presenton the skinof fast swimming sharks. Recent studies by Dean and Bhushan [7] focused on measurements on surfaces emulating the texture of shark skin together with the addition of mucus within the grooves. They postulated that the mucus presentsan added benefit.Further insightinto the physi-caldragreductionmechanismwasprovidedanalyticallybyLuchini etal.[14].The review ofKarniadakisandChoi [13] described ri-bletsasfencesalignedwiththestreamwisedirectionwhichhinder the spanwise propagationof the coherent longitudinalstructures during the‘sweep’event, wherethehighmomentumfluidmoves towards thewall asaresultofturbulent mixing,leading toa re-ductioninthewallshearstress.

Most of these studies have dwelled mainly on riblets, how-ever Tani [17] suggested that randomly distributed roughness could alsoleadtosimilardrag reductionbenefitwithout explain-ing the mechanism in detail. Later, Sirovich and Karlsson [16] demonstrated that a staggered array of discrete protrusions with alambda-shapedcross-sectioncouldalsoreduceskinfictiondrag, whereas alignedprotrusions ledtoadrag increment.They postu-latedthat the physicalmechanismresponsible fordrag reduction was completelydifferenttothatofribletsbutcouldbesimilar to themechanismthatexistsduring spanwisewalloscillationwhich modifiesthephaseofobliquelypropagatingstructures,alteringthe net downwash ofhighmomentum fluid.A morerecent studyby [18] showedthatroundededgesdimplescanalsohelpinreducing skinfrictiondrag.Intheircase,eveniftheReynoldsnumberbased on thediameterwas relatively high, theflow withinthe dimples E-mailaddresses:erwin-ricky.gowree@isae-supaero.fr (E.R. Gowree),

chris.atkin.1@city.ac.uk (C.J. Atkin). https://doi.org/10.1016/j.ast.2018.11.001

Fig. 1. Thetopandsideviewoftheflatplate(left)andtheplatemountedintestsectionofthewindtunnel(topright).Thebottomrightfigureisaschematicrepresentation ofperforatedpaneltoshowthedistributionofthecavities.

wasstillattachedandthedragreductionmechanismwasthrough thegenerationofaconverging-divergingflow.

Thecurrentinvestigation was initially focused on understand-ingthedrag penaltyofstaggereddiscretecircularcavities,as typ-ically found in acoustic liners in turbofanengines, but following unexpecteddragtrends,developedintoastudyoftheflow struc-tureinduced by the cavities which may be analogous to surface protrusioneffectsreportedbySirovichandKarlsson.Theskin fric-tion drag benefit claimed here could potentially be attractive to the aerospace and general transport industry, and for pipe flow andchannelflowapplications.

2. Theexperiment

TheexperimentwasconductedintheT2lowspeedwind tun-nel atthe Handley Page laboratory at City, University of London whichhasaworkingsection of0

.

8 m

×

1

.

12 m

×

1

.

8 m. The ex-perimentalboundarylayer was generatedon awooden flat plate modelwithanellipticalleadingedge,ofratio5:1.Themodelwas also equipped with a trailing edge flap to control the pressure gradient over the plate. The initial laminar boundary layer was trippedusinga0.2 mmtrip-wireplaced at100 mmdownstream oftheleadingedgeanditseffectivenessingeneratingaturbulent boundarylayer was confirmed by the hot wiresignal. A stream-wiserowofsurfacepressuretappings,offsetby100 mmfromthe modelcentreline,wasusedtocheckthepressuregradient. Embed-dedinthe platewas a square cavity ofside 192 mm anddepth 18 mm.Thecavitywasrecessedtoaccommodateasetofa perfo-ratedacrylicpanelssuch thatthesurface ofthepanelswas flush withthesurface of theflat plate. The thicknessofthe panels,h, andperforation spacing, s, were constant at 3 mm and 20 mm respectivelyand the diameterof the perforations, d,varied from 2 mmto5 mm.TherigisshowninFig.1,whichalsoshowsthe surface mounted,single axis traversesystem usedto capturethe velocityprofiles of theboundary layerusing hot wire anemome-try.

King’s law was applied for the hot wire calibration and the velocity profilewas captured usingthe surface mounted traverse whichhasaresolutionof5 μmperstep,togetherwiththe digital-opticalsystem,developedbyGowreeetal.[10],topositionthehot

wireveryclosetothesurfaceoftheplate.Thecombinationofthis traverse probeandtheoptical alignmentsystemhasbeenshown to deliver very accurate valuesfor boundarylayer integral quan-tities.As thetraverse systemhadtobe permanentlyfixed to the surface ofthe model,thehot wiretraverse was limitedto a sin-glelocationalongthecentrelineofthemodelanddownstreamof thefinalrowofthecavities,atx

=

730 mmfromtheleadingedge. Thiscorrespondsto20 mmdownstreamofthelastrowofcavities. From Fig.2, the centreline passesover the centre of the middle rowofcavitiesandthereforecarewastakensothatthe measure-mentstationwasnotinthenearwakeofthecavity.Fromtheflow visualisation in Fig.9,at afreestream speed of15 m/s the near-field,large structuresseemed tobe dissipatedatapproximately3 diameters downstreamof thecavityandtherefore a recovery re-gionof20 mmdownstreamofthewholewaschosen.

Toexplorethestructure oftheflow alongthespanwiseplane, three-componentLDAmeasurements wereperformedusinga tri-axistraversesystemwhichassistedinthemeasurementofalarger spatialdomainthancouldbecapturedbythehotwire.ADANTEC DynamicsLDAsystemwasusedwhileoperatinginbackward scat-termode.Basedonthefocallengthofthelenses,aprobevolume ofapproximately120

×

120

×

2600 μmwasobtained.Allthe mea-surementsweremadeinthetransverseandwall-normalplanes,y andz respectively,atx

=

730 mm.Thiswasapproximately20 mm downstreamofthelast rowofcavities asshownschematically in Fig.2.Thecentrelineoftheplatecorrespondsto y

=

0 mm which was also the centreline of the middle row of holes. Surface oil flow visualisation was also performedto obtaina qualitative un-derstandingoftheflowphysicsinthevicinityanddownstreamof thecavities.The experimentwas conductedwiththeperforations firstsealedonthereverseside oftheacrylic sheets,asshownon thebottom rightofFig.1,andsecondly opento asingle plenum underneaththeperforatedpanel.

Thedesiredzeropressuregradientconditionwithin the exper-imental domain was first confirmed using an unperforated plate over the cavity. Fig. 3 shows the development of the freestream velocityinthestreamwisedirection.Thisresultconfirmedthatthe x

=

730 mm measurement stationwas inthezeropressure gradi-entregion.

Fig. 2. Schematic representation of the top-view of the LDA measurement station.

Fig. 3. The streamwise velocity gradient along the flat plate.

3. Results

3.1. Unperforatedplate

Theinitialhotwiretraverse wasmadeonthebaseline, unper-forated panelforlater comparisonwiththeresults fromthe per-forated cases. From the velocity profiles capturedon the smooth surfacetheskinfrictionwasdeterminedusingClauser’schart tech-nique[3],forwhichthelog-lawwas definedusinga vonKarman constantof0.41.

Theresultingshearstresswasusedtoexpressthevelocity pro-filesinwallunitsasshowninFig.4wheretheprofilescanbeseen

to be consistent with the above choice of von Karman constant andthestandardy-intercept of5.1.From thevelocityprofiles the boundary layer integral quantities were calculated, including the skinfrictioncoefficientwhichisplottedagainstReynoldsnumber, based onthemomentum thickness,

θ

,inFig.5.Notethat dueto theproximity ofthehot wirewiththesurface thefirstfew mea-surementswerepronetoheattransfereffectsandwerediscarded during the calculation ofthe momentumthickness. Although the mostobviouspointswerenotincluded intheprofile,inFig.4this effectisstillseeninthefirstmeasurementpoints.Theresultsfrom the smooth case are compared with the semi-empirical relation givenbyGaudet [8] andtheDNSresultsfromJimenezetal.[12]. At lower Rθ the current experimental resultsshow closer agree-mentwiththeDNSresults;however,throughoutthewholerange of Rθ tested,theagreementwaswithin2%ofGaudet’sresults.

3.2. Perforatedplate

The localskin friction on the perforated panel walls was de-terminedusingClauser’s[6] charttechniqueforthesamevalueof von Karman constant. These results are also presented in Fig. 5. The obtainedskin friction coefficientswere then usedto express the velocity profiles in wall units as shown in Fig. 6. For cavi-tieswithd

>

3 mm,thedownwardshiftinthelogarithmicprofile, showsanincreaseinthelocalskinfrictionsimilartothebehaviour noticed in turbulent boundary layers over rough walls. This be-haviour is in line with the observations of Nikuradse [15] and Clauser[6] and manyothers,summarised by Jimenez[11]. How-ever, the upward shiftin theprofile forthe d

≤

3 mm case sug-gests that a reduction in local skin friction is encountered. This trend is also clearin Fig. 5, which also showsthat the effect of removingthesealundertheperforationsissmall.

From dimensional analysis, the ratio d

/

h and the Reynolds number based on the diameter of the cavity, Rd,are considered to bethe influentialgeometricalparameters andtheskinfriction obtainedfromClauser’stechnique isplottedagainstthese dimen-sionless groups in Fig. 7. From the left hand side of Fig. 7 a well-defined trendis present,suggesting thatford

/

h

≤

1, adrag reductionofupto10%couldbeobtained,whileford

/

h

>

1 there isacorrespondinglysharpdragrise.However,therighthandside ofFig.7suggestthatthedependenceonReynoldsnumberisweak. The accuracy and reliability of Clauser’s technique in the pres-ence of the cavities are questionable due to the modification of the characteristic of the wall in the presence of coherent struc-turesgeneratedbythecavities.Thereforeotherphysicalquantities oftheboundarylayerwhichiscorrelatedtotheskinfrictiondrag werefurtherinvestigated.

Fig. 4. Velocityprofilesalongunperforatedplateatx=730 mm.Theuniversalloglawisdefinedbyz+= (1/0.41)ln u++5.1.(Forinterpretationofthecoloursinthe figure(s),thereaderisreferredtothewebversionofthisarticle.)

Fig. 5. Thelocalskinfrictiononthedifferentofsurfacestested,wherethebaselinesmoothcaseiscomparedwiththeDNSstudyofJimenez[7] andthesemi-empirical relationgivenbyGaudet[8].

Fig. 6. Velocityprofilesaboveperforatedplateatx=730 mm,forafreestreamvelocityof15 m/sandthecavitiessealedatthebottom.Theuniversalloglawisdefinedby

z+= (1/0.41)ln u++5.1.

Fig. 8. The percentage difference in momentum thickness,θ, between the perforated and smooth cases as a function of the ratio, d/h (left) and Rd(right).

Fig. 9. Thenormalisedrmsofthefluctuatingstreamwisecomponent,u/Ue,capturedbythehotwireatafrestreamvelocityof15 m/s(left)and25 m/s(right),forthe

smoothcaseandtheperforatedcasesofd=2 mm andd=5 mm.

Forazeropressuregradientflow,themomentumintegral equa-tioncanbereducedto

d

θ

dx

=

cf

2 (1)

It can be extended to a case of spanswise periodic mean flow based on Ashill and Smith’s [1] formulation of the 3D momen-tumintegral equations andthe assumptionpresented in the Ap-pendixA.

Sothe effectofthe cavities ontheturbulent skinfriction can alsobe quantifiedby consideringthe momentumthickness,

θ

, as presentedinFig.8.Thechangeinmomentumthicknessisa mea-sureoftheaveragechangein skinfrictionover thelengthofthe plate,incontrasttothevelocityprofiles whichindicatelocalskin friction values.

θ/θ

is seento increase withincreasingd

/

h and buttherateofchangeislesssteepthan



cf

/

cf (Fig.7).Moreover, forthelowestspeed tested, U

=

15 m/s,areduction in

θ

canbe observedforall theratios ofd

/

h tested.Thisobservation contra-dictsthetrendfromFig.6and7,whereaskinfrictionpenaltywas observed forthecases ofd

>

3 mm. Therelation between

θ/θ

and Rd is more consistent and there seems to exist a particular regimeatRd

<

6000 wheretheskinfrictionreductionisobserved. InFig.8ananomalousdatapointforthecaseofU

=

20 m/s,with sealedholeshasbeenomitted.

So far the skin friction drag has been treated purely from a meanflow basis.Furtherevidencecouldbeobtainedbyanalysing

the RMS of the fluctuating components and from the hot-wire measurements the streamwise component, u, was captured and ispresentedinFig.9.Atafreestreamvelocityof15 m/s,thepeaks in u for 2 mm and 5 mm cavities are slightly lower than that of the smooth case, this therefore confirms that skin friction is lower for these cases. As opposed to the case of 25 m/s where thepeakofthe5 mmcavitieswashigherthanthe2 mmandthe smooth cases,whichwereinturnvery similar.It isalso interest-ing to note that at15 m/s, inthe outer region of the boundary layer, y

/δ >

0

.

6, the intensity of u in the presence of the cavi-ties is lower than the smooth case, whereas at 25 m/sthey are all quite similar. In the inner layer region, y

≈

0

.

2, forthe cavi-ties, d

=

2 mm,andsmoothcaseu areverysimilarwhereas that ford

=

5 mm ishigher.Withincreasing freestreamspeed the in-tensityinu,shouldincreasefurtherandwillmergewiththepeak intoaplateausimilartothecaseofverylargeroughness. 3.3. Topologyofsurfacestreamlines

Thesurfaceoilflowpatternsinthevicinityanddownstreamof the cavities are shown belowin Fig. 10.The streamline patterns areconsistentwiththeformationofvorticalstructuresaroundthe cavities. For the three cases on the upper left of Fig. 10 a drag reduction was observedwhereas forthethreecases tothelower rightadragpenaltywasobserved.Thedevelopmentofthe stream-linepatternsasholesizeandfreestreamvelocity areincreasedis

Fig. 10. Surfacestreamlinevisualisationoftheflowaroundthecircularcavityofd=3 mm(left)andd=5 mm(right).Wherethefreestreamvelocityfromtoptobottom arefrom15,20,25 m/srespectively.

Fig. 11. The normalised mean streamwise velocity component, u/Uefor the d=3 mm (left) and d=5 mm (right), with Rd≈5000 and Rd≈8500 respectively. rathersubtle,butforthe casewhere thegreatest drag reduction

wasobserved(d

=

3 mm,at15 m/s)thereareclearsignsoflateral velocitiescomponentinthesurfacestreamline.

3.4.3DLDAmeasurements

The LDA data was filtered so as to discard all measurements witha validation level less 30%. A spline fit algorithm was em-ployed to interpolate the velocity contours, resulting in a more organised representation of the turbulence structures. The

nor-malisedstreamwise velocitycomponentsareshowninFig.11for the cases of d

=

3 mm on the left and d

=

5 mm on the right, forRd

≈

5000 andRd

≈

8500 respectivelyatU∞

≈

22

.

5 m/s.Note that atthisReynolds number,a drag reduction was observed for d

=

3 mm whereas adragpenalty was observedford

=

5 mm in comparisontoa smoothsurface. Thestreamwise velocity compo-nents showninFig. 11reveal aregion oflower momentumflow nearthewallat y

= ±

5 mm.AsshowninFig.2theholecentres lie at y

=

0 mm and y

±

10 mm. Thepockets oflow streamwise momentumfluidarethereforeconsistentwithasecondaryflow

in-Fig. 12. Thenormalisedrmsstreamwisevelocity,u/U e (top),transversev/U e (centre)velocitycomponentsandbottom(u2−v2)/U2

e,forthed=3 mm (left)andd=5 mm

Fig. 13. VelocityprofilesexpressedasClauserchart,forthreepositionalongthespanwiseplanelocatedatx=730 mm,forthecaseofd=3 mm(left)andd=5 mm(right) withRd≈5000 andRd≈8500 respectively.Rz,representstheReynoldsnumberbasedonthelocalheight,z,withrespecttothewall.

ducedbytheadjacentlegsoftheneighbouringholevortices.The sense of rotation is to lift low momentum fluid in between the holesandto bringdown highmomentum fluidbehind the holes whichisconsistentwiththeoilflowpatternsseenatthetopleft ofFig. 10. The rms of the fluctuatingstreamwise and transverse velocitycomponents are shownatthetop andbottom ofFig.12 respectively, ford

=

3 mm (left) andd

=

5 mm (right). The sec-ondary flow from the hole vortices appears to lift the turbulent nearwall flow away from the surface for the caseof d

=

3 mm butnot ford

=

5 mm. Thisis consistent withthe observed drag reductionford

=

3 mm.

4. Discussion

The skin friction measured for the baseline smooth surface agrees well with the universal ‘log-law’, the skin friction results obtainedfromDNSandsemi-empiricalresults,allshowninFig.4. Thisprovides confidence in the measurement technique and the dataanalysisprocess.Theuncertainty intheskinfrictiondrag in-ferredfromtheClauser’stechniqueandthemomentumdeficit,d

θ

relyontheaccuracyofthevelocityprofilesmeasurements.Gowree etal.[10] showedthatthetraversemechanismcouldcomfortably providea singlestep displacementof5 μm atan accuracy lower than

±

5%.FurthererroranalysisbyGowree[9] demonstratedthat theopticalalignmentallowedthepositioningofthehot-wirewith an accuracy of

±

2.5 μm with respect to the wall and within a confidencelevelof95%,thewholeboundarylayertraversecanbe capturedata total relative uncertaintyof 3.9%.This was for tur-bulentvelocity profiles with a maximum thickness of 3 mm, so hereitwillbe fairto assumethattheconfidencelevelinthe ve-locityprofile measurement is even higher and therefore a lower totalrelative uncertaintycan be expectedastheboundary layers are about 4 to 5 times thicker. From Fig. 6 the upward shift of the inner layer for the caseswhere d

<

3 mm suggests that the local skin friction is reduced. This is supported by the fluctuat-ing velocity componentspresented inFig. 12.We do not believe thatthisobservedskinfrictionreductionoversome typesof per-foratedsurfaceshasbeenreportedpreviously.Smalldifferencesin skinfriction betweentheopenedandsealed cavitiesare thought tobeduetolowlevels ofpressurefluctuation,resultinginforced momentumtransfer into theplenumunder theperforated sheet. Larger magnitudesofunsteady transpiration(e.g.inthe presence ofstrongacousticforcing)wouldbeexpectedtoleadtolargeskin

frictionincrementswhichwereclearlynotobservedinthepresent work. UsingtheClauserchart technique,a skinfrictionreduction was observedforacavitydiametertoheightratio,d

/

h

<

1,buta differenttrendwasobtainedfromthemomentumthicknessresults which suggesteddrag reduction for all Rd

<

6000.We note here thattheClausercharttechniquemightberestrictedtocaseswhere the log-law constants do not differ from the equivalent smooth wall values. From the hot wire measurements the peaks in the intensityofthefluctuatingstreamwisecomponent,u,forthe cav-itiesinthisregimewerelowerthantherelativesmoothcasesand thisalsosupportsthedragreductionobservedhere.

Surfaceflowvisualisations(Fig.10)revealedthespanwise non-uniformity of the flow and the presence of transverse velocity componentsandstreamlinepatternsconsistentwithvortical struc-turesgeneratedbythecavities.Theeffectofthespanwisevariation onthelocalskinfrictionwascheckedbyplottingthevelocity pro-filesfromtheLDAmeasurements ona Clauserchart asshownin Fig. 13. The LDAvelocity profiles suggest that thespanwise vari-ation in skin friction was small andsymmetrical suggestingthat drag reductionsextended over the whole span of the perforated plateandisnotjustalocalisedbenefitobservableinthewakeof thetrailingvortex.Wenotethatforthecaseofd

=

5 mm shown ontherightofFig.13,nosignificantvariationcanbeseenbetween theprofilesatthethreespanwisepositions.

TheLDAmeasurementswereusefulinexplainingthefootprint of the surface streamlineshown in Fig. 10 and in capturing the effecton theboundary layer turbulence,offering insightintothe possible mechanism of drag reduction. Referring back to the In-troduction, in previous studies skin friction reduction had been achieved by manipulating the transverse velocity component ei-therbyusingriblets,spanwiseoscillationorrandomlydistributed protrusions.AccordingtoBerchertetal.[6],areductioninthe fluc-tuatingtransverseorspanwisecomponentnearthewallwillresult in a reduction in turbulent energy production and hence lower shearstress.Withthecurrentlimitedresultsonecanonly specu-latethatthereducedtransversevelocityfluctuationsaty

±

5 mm forthe d

=

3 mm case could be a contributing factorto the ob-served skin friction reduction. These features were not visible in thed

=

5 mm case.Ford

=

3 mm boththemeanstreamwiseand fluctuatingtransversevelocitycomponentsareorganisedina sim-ilar fashion to that of the flow over riblets, suggesting that the dragreductionmechanismcouldbesimilar.However,thescenario present in the case studied by Sirovich and Karlsson cannot be

ruled out. For the d

=

5 mm case Fig. 12 shows the transverse velocity fluctuationstobe higherinmagnitude andfurtheraway from the wall than the streamwise velocity fluctuations which isperhaps associatedwithunsteadiness inthe vorticalstructures whichmayexplain the Reynoldsnumberdependenceof the sec-ondaryflow.

5. Conclusion

Measurementsofthelowspeedflowfieldaroundstaggered ar-ray of three dimensional cavities have demonstrated a potential passivemethodforskinfrictiondrag reduction.Followinga para-metricstudy,thedragbenefitwasobservedtobelimitedtosome extentbytheratiod

/

h andtobeastrongfunctionoftheReynolds numberbasedonthediameterofthecavity,Rd.Surfaceflow visu-alisationandLDAmeasurementsacrossaspanwiseplaneindicated thepresenceofvortices shedby thecavitieswhich,forthecases exhibitingreducedskinfriction,organisethetransversefluctuating velocityinasimilar fashiontoriblets.Abenefitofstaggered cav-itieswouldbealowerexcrescencedragpenalty incomparisonto thatfromriblets.

Furtherworkshouldexploreingreaterdetailthedevelopment oftheflowstructureswithReynoldsnumberand/orcavityspacing andgeometrytounderstandthelimitationonthemechanism. Conflictofintereststatement

Theauthorsdeclareisnoconflictofinterest. Acknowledgements

TheauthorswouldlikethanktheInnovateUKfortheirfinancial supporttotheSANTANAprojectundergrantNo.113001.

Appendix A

Theanalysisapplicablehereisfortheboundarylayerflow un-dera straightstreamlineina zero-pressure-gradient, incompress-ible, three-dimensionalflow: a rare occurrence, but one example would be the viscous layer between a two-dimensional inviscid flowandathree-dimensionalsurface,suchastheperforatedplate of the current study. The three-dimensional momentum integral equationscanbeobtainedfromanumberofsources,butherewe follow Smith and Ashill (RAE TR 83053) and cite only the mo-mentum integral equation in the streamwise direction, modified toallow forzeropressuregradient,incompressibleflow andzero streamlinecurvature:

∂θ

11

∂

s

+

∂θ

12

∂

n

=

τ

w1

ρ

U2e (2)

Heres andn arerespectivelythedirectionsparallelto,and nor-malto,the externalstreamline; Ue is theinviscid velocityatthe boundarylayeredge;

ρ

isthedensityoftheincompressiblefluid; and

τ

w1 is the wall shear stress inthe direction ofthe external streamline.Theintegralscalesaredefinedby:

θ

11

=

e



w u Ue



1

−

u Ue



dz (3)

θ

11

=

e



w v Ue



1

−

u Ue



dz (4)

where u and v are respectively the velocity component inthe s andn directions andw ande respectively indicate conditions at

the wall and at the boundary layer edge. Additional terms are included by Ashill and Smith arising from streamwise pressure gradient,fromstreamlinecurvatureintheinviscid flowandfrom gradients ofdensity,all ofwhichareabsent inthepresent situa-tion.

Inthecontrolvolumeanalysiswhichcanbeusedtoderiveboth thedifferentialandintegralboundarylayerequations,theterm

∂

∂

n

⎡

⎣

e



w v Ue



1

−

u Ue



dz

⎤

⎦

(5)

represents the net change in streamwise momentum due to the gradient in lateral convection of streamwise momentum across the sides of the control volume. Any net outflow of streamwise momentum due to the normal (crossflow) velocity component, v, would result in an inflow of momentum to the neighbouring control volume and vice versa. Thus, for a flow which is two-dimensional from the macroscopic perspective, anyimbalance in lateral flow of momentum would diminish in importance as the control volume was extendedacross theflowfield. In thepresent context, the significance of thiseffect can be determined by the variationofmomentumthickness,

θ

11,inthelateraldirection(i.e.

across thestream).Assumingthatthisvariationisvery small,we canthenapplythetwo-dimensionalformofthemomentum inte-gralequation

∂θ

11

∂

s

=

τ

w1

ρ

U2e (6)

effectively treating the localised effects of the three-dimensional surface features asamodifier tothemean,two-dimensionalwall shearstress,

θ

11.

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