Quantitative criteria to design optimal permeable trailing edges for noise abatement

Quantitative criteria to design optimal permeable trailing edges for noise abatement

Rubio Carpio, Alejandro; Avallone, Francesco; Ragni, Daniele; Snellen, Mirjam; van der Zwaag, Sybrand

DOI

10.1016/j.jsv.2020.115596

Publication date

2020

Document Version

Final published version

Published in

Journal of Sound and Vibration

Citation (APA)

Rubio Carpio, A., Avallone, F., Ragni, D., Snellen, M., & van der Zwaag, S. (2020). Quantitative criteria to

design optimal permeable trailing edges for noise abatement. Journal of Sound and Vibration, 485,

[115596]. https://doi.org/10.1016/j.jsv.2020.115596

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ContentslistsavailableatScienceDirect

Journal

of

Sound

and

Vibration

journalhomepage:www.elsevier.com/locate/jsv

Quantitative

criteria

to

design

optimal

permeable

trailing

edges

for

noise

abatement

Alejandro

Rubio

Carpio

∗

,

Francesco

Avallone,

Daniele

Ragni,

Mirjam

Snellen,

Sybrand

van

der

Zwaag

Faculty of Aerospace, Delft University of Technology, Delft 2629HS, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 9 December 2019 Revised 28 May 2020 Accepted 20 July 2020 Available online 21 July 2020

2019 MSC: 00-01 99-00

Keywords:

Trailing edge noise Noise control Porous materials

a

b

s

t

r

a

c

t

An experimental study on broadband noise scattered by permeable trailing edges with dif- ferent pore arrangements is performed. A NACA 0018 airfoil with chord c = 0.2 m is in- vestigated at chord-based Reynolds numbers ranging from 1.4 × 10 5_{to 3.8 × 10} 5 _and

angles of attack of 0.2 and 5.4 degrees. Noise emission from five 3D-printed perforated trailing-edge inserts, with channels normal to the chord, is measured with a microphone antenna. For comparison, inserts manufactured with metallic foams, with comparable flow permeability K but more tortuous pore paths, are also analysed. All the inserts have a per- meable extension s equal to 20% of the chord ( s/ c = 0.2). It is shown that noise mitigation

Lp, computed as the difference between far-field noise scattering from solid and perme-

able edges, collapse when nondimensionalizing frequency as Strouhal number based on the chord. From the collapsed data, it is observed that the maximum noise attenuation reported for each insert Lp,max reaches an asymptotic value of 9.3 dB for increasing K.

To parameterize such asymptotic behaviour, noise reduction levels are fitted to a newly proposed relation Lp,max = γ1tanh (γ2K) (where γ1 and γ2 are fitting coefficients, that

depend on the type of insert and angle of attack). Following this analysis, limit permeabil- ity values for perforated and metal foam inserts of K = 3.5 × 10 −9_{and 1 × 10} −9_m 2_are

found, respectively; above these thresholds, less than 1 dB additional noise mitigation is reported; below, a difference in Lp,maxof up to 4 dB for a given Kis measured depending

on the pore organization. Consequently, the tortuosity of the permeable structure is iden- tified as an additional parameter (to K) controlling noise attenuation. It is also observed that the acoustic performance of lower-permeability edges is less sensitive to changes in the angle of attack. Tests for permeable lengths equal to s/ c = 0.05 and 0.1 are performed: the change of Lp,maxwith increasing s/ c is also properly described with a hyperbolic tan-

gent, evidencing equally good performance in noise reduction for all measured extents. Finally, for the most permeable insert with periodic pore arrangement, an extremely loud tonal noise caused by vortex shedding (+30 dB higher than broadband levels) from the blunt solid-permeable junction at s/c = 0 .8 is reported. Since applying a longer permeable surface, or increasing the permeability at the trailing edge decreases the aerodynamic per- formance of the blade, a permeable trailing edge with s/ c = 0.05, K = 1 × 10 −9 _m 2

and tortuosity of 1.15 is recommended to optimize broadband noise abatement and avoid shedding-related tones for the conditions explored in the current study.

© 2020 The Authors. Published by Elsevier Ltd.

∗ _{Corresponding author.}

E-mail address: a.rubiocarpio@tudelft.nl (A. Rubio Carpio). https://doi.org/10.1016/j.jsv.2020.115596

0022-460X/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Turbulent-boundary-layertrailing-edgenoiseisasignificantcontributortonoisegeneratedbywindturbines[1], ventila-tionsystems[2]oraircraftairframes[3].Thisnoisesourceisproducedbyturbulenteddiesbeingdiffractedattheaftedge ofliftingdevices[4].Theemploymentofpermeabletrailingedges canmitigatenoiseupto25dB[5]withrespecttotheir equivalentsolidconfiguration[6].Itisnowwell-establishedthatpermeabletrailingedges mitigatetheacousticimpedance jumpattheedge,andpromotenoisescatteringfromdifferentstreamwisepositions [7–9].Suchaprocess iscontrolledby thelocal permeabilityof thematerial[10],that determinesthe degreeofflow communicationbetweenbothsidesofthe airfoil(thusthemagnitudeoftheacousticimpedancejump).Consequently,theemploymentofmorepermeablestructures usuallyyieldslargernoisereduction[11].

Previous research on permeabletrailing edges employed (open-cell) porous structures such asfoams [12,13], sintered granulates[10], fiberfelts[5] ormetal meshes [14]. Materialswith such a complexpore arrangement oftenentail addi-tional fluid-dynamicand acoustic features. Forinstance, hydrodynamically rough surfaces produce high-frequency excess noiseandincreasedturbulencelevelsclosetothewall[15];theseeffectsmightpartiallymasktheiracousticbenefits[9,11]. Furtheraspects, such asthechangeofpropertieswhen subjecttoconventional machiningprocesses,orrepeatability and inhomogeneityissuesalsodiscouragetheir useforresearch.Recentlydevelopedmanufacturingtechniquessuchasadditive manufacturing (or 3D-printing) [16] allow constructing permeabletrailing edges withchannels that connect suction and pressuresidesoftheairfoil[17].Consequently,edgeswithtailoredpermeabilitycanbemanufactured byvaryingholesize, shapeorspatialdensity.

Inthecurrentmanuscript,fivedifferentpermeabletrailingedgesforaNACA0018areacousticallycharacterizedatangles ofattack

α

of0.2and5.4degreesandchord-basedReynoldsnumbersRe_c rangingbetween1.8_{× 10}5 _{and4.5× 10}5_.These

permeableedges,withstraightchannelsnormaltothechord,allowforflowcommunicationbetweensuctionandpressure sidealongthelast20%oftheairfoil.Thefiveperforatedinsertshavethesameholediameterd_h butdifferentpermeability K,obtained by varying the spacingbetween holes l_h . Noise scatteringof such perforated trailing edges, with a periodic channel arrangement, is compared to that of open-cell metal foam inserts, with different (random) micro-structure but similar macroscopicpermeability. The comparisonyields generalcriteriathat might serveas guidelinesforthe designof futurepermeabletrailingedges.

Thecurrentpublicationisorganizedasfollows.Theexperimentalset-up,the3D-printedinsertsandtheacousticphased arrayaredescribedinSection2.Then,far-fieldnoiseresultsarediscussedinfivesubsections:firstly,scalinglawsfornoise mitigation withflowspeed ormaterialpropertiesarepresentedinSection3.1;thetortuosity ofapermeableedgeis pro-posedto furthercontrol andoptimizenoiseabatement inSection3.2; thescalingof far-fieldacousticpressurewithflow speedisreportedinSection3.3;abriefanalysisontheeffectofvaryingthechordwisepermeablelengthonnoise mitiga-tionispresentedinSection3.4;finally,theoriginofatonemeasuredforthemostpermeableperforatedinsertisaddressed inSection3.5.Toconclude,inSection4asummaryofthemainfindingsispresented.

2. Experimentalset-up 2.1. Windtunnelandmodel

Experimentsarecarriedoutintheaeroacousticverticalopen-jetwindtunnel(A-Tunnel)atDelftUniversityof Technol-ogy.The rectangularcross-sectionatthenozzleoutletis0.4 × 0.7 m2 _{(contraction ratioequalto15:1). Forthepresent}

investigation,experimentsare performedatfree-stream velocitiesof12.5,15,20,25,30and34m/s. Inflowconditions at the exitsection are characterizedby means ofa Pitottubeandconstant temperatureanemometry (Dantec DynamicsP11 probe; TSIIFA300 bridge).The streamwisevelocity isuniformwithin 0.6%atthe measuredcross-section,andthe turbu-lenceintensityisbelow0.06%fortherangeoffree-streamvelocitiesspecifiedabove.

A modularNACA 0018airfoil withchord c ₌ 200 mm and spanL ₌ 400 mm(Fig. 1(a–c))is placed atthe potential coreofthejetwiththeleadingedge500mmdownstreamthenozzleoutlet. Themodelisinstalledbetweentwowooden sideplates(withheightequalto1.2m)flush-mountedtothenozzleoutlet(Fig.1(d)).Thehighspan-to-chordratioL/c=2 suggeststhatthree-dimensionalfloweffectsarenegligible alongthemidspanline;thisaspectisfurtherconfirmedbythe goodagreementbetweenexperimental boundarylayerdataobtainedwiththesameset-up [9],XFOILdata[18]and high-fidelitysimulationsonatwo-dimensionalmodel[19].

The aluminium body of the airfoilis manufactured withnumerical control machining (surface roughness: 0.05 mm). Thelast20%oftheairfoilcanbereplacedbypermeableinsertsmadeofdifferentmaterialsandpermeabilitypatterns.The insertsaresimplyattachedtotheairfoilbyadovetailconstruction,andtheirsurfaceisflush tothesolid airfoilsurface.A piece ofScotchCrystal Tape(width: 19 mm;nominal thickness:51 μm) iscarefullyapplied atthe junctionbetweenthe

Fig. 1. Experimental set-up. The aluminium body of the airfoil is depicted in light grey. The exchangeable trailing-edge insert in dark grey. (a) 3-dimensional isometric view. (b) Side view. (c) Front view. (d) Picture of model and side plates assembled on the nozzle.

solidbodyoftheairfoilandthepermeable/solidtrailing-edgeinsertsinsuchawaythatthetapecovers1mmoftheinsert inthestreamwisedirection.Thisprocedureensuresasmoothtransitionbetweenbothpieces.Thetrailing-edgeinsertshave athicknessof15.7mmattherootand0.3mmatthetip.

Boundary-layertransitionto turbulenceisforcedat20% ofthe chordon both sidesoftheairfoil. The trippingdevice, withachordwiseextentof10mm,has0.84mm-heightcarborundumelementsrandomlydistributedalongtheentirespan. The turbulent state ofthe boundary layer is confirmedby scanning the airfoilsurface with a Brüel& Kjaer(B&K) 4134 microphoneconnectedtoaB&K2619preamplifier.

Thestreamwise-vertical-spanwise X − Y − Z coordinate system, employed inthe remainingof themanuscript,is also depictedinFigs.1(a–c). Itsoriginislocatedattheintersectionbetweenthemidspanplaneandthetrailingedge,XandZ axesarerespectivelyalignedwithchordandtrailingedge,andtheYaxispointstothemicrophoneantenna.

2.2.3D-Printedtrailingedges

Channelled trailing-edge inserts are manufactured withan EnvisionTEC’s Perfactory 4Standard (resolution:25 μm), a DigitalLightProjector(DLP)printer.Theinserts(surfaceroughness:0.03mm)areprintedfromroottotipusingEnvisionTEC’s HTM140 V2,a high-temperature photopolymer [20]. The totalchordwise length ofthe inserts is 60mm, ofwhich only s/c=0.2=40mmareexposedtotheflow(Fig.1(b)).Themaximumspanofeachinsertislimitedbythebuildenvelope oftheprinter(160 _{× 100× 180}mm3_{)to100mm;hence,4insertsareassembledtobuildtheentiretrailingedge.}

The inserts containstraight cylindricalchannels normalto the chord connecting suction andpressure side. Holesare distributedaccordingtoastaggeredsquarelattice,asshowninFig.2.Inthepresentinvestigation,d_h isequalto0.8mmto ensurethatthechannelsareopen aftertheprintingprocessandtoavoidlow-frequencyacoustictones[11].Theflow per-meabilityKoftheinsertsisthusvariedbychangingtheholespacingl_h .Fivedifferentperforatedinserts,withhomogeneous holespacingl_h of1.5,2,2.5,3and5mm,arestudiedinthepresentmanuscript.Holespacingsarechoseninordertoobtain perforatededgeswithsimilarpermeabilitytothosemanufacturedwithmetallicfoams(thepermeabilitycharacterizationis discussed morein detail later in Section 2.3). The smallest holespacing (lh = 1.5mm)is determined by manufacturing

constraintstoguarantee0.3mmofsolid materialbetweenorifices,whichyields asatisfactoryoutcomeofthe3D-printing process.Picturesoftheinsertswithl_h =1.5and5mmareshowninFig.3(a)and(b)respectively.

Fig. 2. Sketch of the hole pattern for perforated inserts.

Fig. 3. Pictures of the 3D-printed trailing edges. Total chordwise length is 60 mm (of which exposed to the flow s / c = 0.2 = 40 mm). Total span is 100 mm. (a) l h = 1.5 mm. (b) l h = 5 mm. Note that a wavy edge results from the l h = 1.5 mm hole distribution. A similar feature can be found in mock-ups with

lh = 2, 2.5 and 3 mm..

After the printing process it is observed that holes are not perfectly round due to limitations in the manufacturing technique[21];specifically,themostdownstreamwiseendofeach holeisflattened, producingD-shapedchannels.Optical microscopy (Fig. 4) showsthat theseholes have a major diameter of approximately800 μm and a shortenedside with lengthof700μm.Toaccountforthisfeature,anequivalentholediameterof754μm,thatyieldsacirclewithareaequalto themeasuredopenarea(of447000μm2_{),isemployedinSection2.3.}

2.3. Flowpermeabilityandformcoefficient

FlowpermeabilityKandformcoefficientCoftrailing-edgeinsertsaredeterminedbymeasuringthestaticpressuredrop



pacrossperforatedsampleswithsimilarholedistributions. Thepressuredropacrossa sampleofpermeablematerialis determinedbytheHazen-Dupuit-Darcyequation[22]



p

t =

μ

K

ν

+

ρ

C

ν

2 ₍₁₎

where t is the thicknessof the sample,

ρ

is the fluid density,

μ

isthe dynamic viscosityand

ν

is the Darcian velocity (defined as the ratioof volumetric flow rate to samplecross-section area). To characterize the two hydraulic properties (KandC)ofeach holepattern,4cylindricalsampleswiththicknesst=10mm,representativeoftheaveragetrailing-edge insertthickness,aremanufactured.Thesesampleshaveholepatternswithl_{h =}1.5(Fig.5(a)),2.8,4.5and6mm((Fig.5(b))), andanouterdiameterof55mm.ToyieldthesameD-shapedholefeature,samplesareprintedasthetrailing-edgeinserts, i.e., the printing directionis perpendicular to the channels’ axes. K andC are then computedby least-squares fittingof

Fig. 4. Microscope picture of the perforated plate with l h = 1.5 mm employed to characterize the hydraulic properties of the hole pattern. Hole open area

is shadowed in red. Printing layers are distributed horizontally. Printing direction goes from top to bottom. For trailing-edge inserts air flows from top to bottom. Dimension: 1278 × 1114 px 2 . Resolution: 2 μm/px. (For interpretation of the references to colour in this figure legend, the reader is referred to

the web version of this article.)

Fig. 5. Pictures of the 3D-printed cylindrical samples employed to characterize flow permeability and form coefficient of the different hole distributions. (a) l h = 1.5 mm. (b) l h = 6 mm.

Table 1

Fitting coefficients for Eq. (2) (a) and (b).

Study β1 β2 β3 R 2

Bae and Kim [24] 32 15 0.75 0.98 Present 40 15 0.73 0.95

HydraulicpropertiesfortheholedistributionsemployedinthetrailingedgesarethenestimatedusingEq.(2)(a)and(b),

K=

σ

d 2 h t

β

1t+

β

2dh (2a) C=

β

3 1−

σ

σ

2_t (2b) where

σ

=

π

d2

h/

(

2lh2

)

istheporosity,definedastheratiooffluidtosolidvolume.Theseempiricalcorrelationsaredefined

byBaeandKim[24] forperforatedplateswithasimilarholearrangement andfullycylindricalholes.The originalfit co-efficients

β1

,

β2

and

β3

, aswell astheonesretrievedwithpresentdata,are reportedinTable1.Tovalidate theresults, correlationsbasedonmeasureddataarecomparedtotheonespresentedbyBaeandKim[24]inFig.6(a)and(b).For com-pleteness,KandCvaluescomputedfromEq.(1)(referredtoas”Measurements”),andtheonesinterpolatedatl_h employed intrailing-edgeinserts(referredtoas“Interpolation”)areshown.Therelativeerrorfortheoriginalcorrelationsstatedbythe

(a) (b)

Fig. 6. Flow permeability K (a) and form coefficient C (b) as a function of the hole spacing. Measurements (diamond markers) are compared to empirical correlations (dashed line) reported in [24] , together with the 3% relative error stated by the authors (grey shadowed area).

Table 2

Characteristics of permeable micro-structures measured on samples with

t = 0.01 m. P stands for perforated and MF stands for metal foam. K and C values are interpolated from Eq. (2) (a) and (b) employing the corresponding l h

value; measured values (computed from fit of experimental data to Eq. (1) ) were reported in a preliminary analysis [25] , hence the small discrepancies among them. Similarly, the porosity σ is now computed with the corrected d h of 754 μm to

account for D-shaped holes.

Type dc (mm) dh (mm) lh (mm) σ(-) K (m 2 ) C (m −1 ) P - 0.8 1.5 0.40 54 × 10 −10 ₂₇₉ P - 0.8 2 0.22 31 × 10 −10 ₁₁₃₈ P - 0.8 2.5 0.14 20 × 10 −10 ₃₀₆₅ P - 0.8 3 0.10 14 × 10 −10 ₆₆₈₁ P - 0.8 5 0.04 5 × 10 −10 _55,189 MF 1.20 - - - 64 × 10 −10 _- MF 0.80 - - 0.917 32 × 10 −10 ₂₃₃₃ MF 0.58 - - 0.905 15 × 10 −10 ₃₉₃₉ MF 0.45 - - 0.893 5 × 10 −10 _10,335

authors(equalto3%)isalsodepictedasagreyshadowedarea.Resultsshowthatbothcorrelationsareingoodagreement, confirmingthegoodnessofthepermeabilitycharacterization.

Finally,asummaryofthecharacteristicsofallperforatedconfigurationscreatedispresentedinTable2,wherethe resis-tivityR₌K_/

μ

isalsoreportedforcompleteness.Sincethenoisescatteringanalysisincludesacomparisonwithinserts man-ufacturedwithmetalfoams,asummaryofrelevantparameterspreviouslymeasured[9,23]formetal foamswithdc =450, 580and800μm,isalsoincluded.Forthemetalfoamwithdc =1200μm,permeabilitydataasreportedbythemanufacturer isincluded.

2.4. Tortuosity

A preliminarycomparison betweenmetal foams andstraight periodically channelledinserts [25] showedthat trailing edges withdiverseflowpermeabilityandformcoefficient mightabatenoisesimilarly;hence,althoughitiswidely recog-nizedthatthesetwopropertieshaveasignificantimpact[5,11],theymightnotbesufficienttofullycharacterizethenoise mitigation capabilitiesofstructures withdifferentporearrangements.Inthepresentmanuscript,tortuosity

τ

is proposed toaccountfortheverydifferentmicro-structureofmetalfoamandchannelededges.Tortuosity

τ

_{≡ lp} /tisusuallydefined1

astheratiobetweentheaverageporelengthlp tothethicknessoftheporousmediumt[28].Fromthisdefinition,itstems that straightchannelshave

τ

_Y =1.Forthemetallicfoams,thetortuosity oftheporephaseisstatisticallycomputedwith randomwalkmodels[29].Tothisaim,pytrax[30],an open-sourcePythonsuiteisemployed onCT-scannedsamples(size:

1 Note that a distinction between tortuosity _{τ, as defined in the present manuscript, and tortuosity factor κ= τ}2 should be made _{[26] . For more infor-}

Table 3

Tortuosity of metal foams (void phase) computed with random walk methods.

dc (μm) τX τY τZ τ

450 1.142 ± 0.007 1.194 ± 0.007 1.141 ± 0.007 1.158 ± 0.004 580 1.115 ± 0.006 1.153 ± 0.007 1.143 ± 0.006 1.137 ± 0.004 800 1.107 ± 0.006 1.129 ± 0.006 1.113 ± 0.006 1.116 ± 0.003

1200 1.079 1.061 1.088 1.076

10 × 10× 5mm3_{;resolution:12μm)ofmetalfoamswithd}

c =450,580and800μm.Randomwalkprocessesmimicthe diffusionprocessofnon-sorbingparticles,i.e.walkers,randomlydistributedwithinporousmedia[31].Thealgorithmbegins atimesequence inwhicheach walkerrandomlyselectsa neighbouringvoxel;ifthechosen voxelisalsovoid,thewalker thenmigratestothatposition;ifnot,thewalker doesnotmoveatthattime-step [27].Themeansquareddisplacementof theensembleofrandomwalkersr

(

t

)

2_{isthencomputedas}

r

(

t

)

2₌ 1 nw n w  i =1



r_X,i

(

t

)

2_+r Y,i

(

t

)

2+rZ,i

(

t

)

2



(3)

wheren_w is thetotal numberofwalkers,andr_X,i ,r_Y,i andr_Z,i referto thedisplacement ofthewalker i along X,Y andZ respectively.The mean squareddisplacement within porous media r

(

t

)

2

p is then lower than theone obtained ina fully

voidspacer

(

t

)

2

v.Thedegreeofreduction,i.e.

τ

,isthencomputedas[32]

τ

=









r2˙ v ˙ r2_p (4)

wherer˙refers tothetimederivativeofr.Anaccurate computationof

τ

isdependantonboththenumberofwalkersand time-stepsnt .Toobtainreliable

τ

values,50,000walkersand50,000time-stepsareemployedafterperformingaparametric study.From Eq.(3), itcan beinferred that consideringonly axialmeansquare displacements, thetortuosity ina specific direction(

τ

_X ,

τ

_Y ,

τ

_Z )canbealsocomputed.Results,presentedinTable3,areobtainedfromanensembleof100simulations; reportedvaluescorrespond totheensemble average,andtheuncertainty iscalculatedemployinga 95% confidencelevel. From theresults,it stems that metal foams are not perfectlyhomogeneous. Tolimit theuncertainty onthe aeroacoustic analysisderivedfromusingmaterialswithsomewhatheterogeneousproperties,thesamebatchoffoammaterialwasused forthetortuositycomputationandtheinsertconstruction.Similarly,thepermeabilitycharacterizationiscarriedoutinsuch awaythatK≡ KY andC≡ CY .

ThetortuosityvaluesofTable3areinlinewiththosepreviouslyreportedintheliterature[33–35]for open-cell metal foams with similar characteristics. Finally, due to the lack of availability of a CT-scan sample for the metal foam with dc =1200μm,itstortuosityisobtainedbyextrapolatinglinearlymetalfoamdatabasedonthecelldiameter.

2.5.Open-jetcorrectionforangleofattack

Experimentsare performed atgeometric angles ofattack of0 and8 degrees. However, the open-jetconfiguration of theexperimentalset-updecreasestheeffectiveangleofattack

α

.Forthisreason,

α

iscomputedbyanalysingthemeasured pressuredistributionalongthechord.Staticpressuremeasurementsareobtainedthrough15differentialpressureHoneywell TruStabilitytransducers(range: ± 2.5 kPa;accuracy:12.5Pa),connectedto 30pressuretaps(with diameterof0.4mm). TapsaredistributedwithinX/c₌-0.99to-0.34alongaplaneinclined15degreesandwithanoffsetof20mmwithrespect tothemidspanplane(Fig.1(a–c)).Thisfeaturepreventsflowinterferencebetweenconsecutiveholes.

Pressuredataaresampledfor15satasamplingrateof50Hz.

α

isdeterminedbycomparingthestaticpressure distri-butionforthereferencecaseagainst XFOILdata[18],whichyieldsresultsforan airfoilinfree-airconditions.Thepressure coefficient,definedasCp =

(

P− P∞

)

/

(

0.5

ρ

U∞2

)

,isplottedasafunctionofthechordwisepositioninFig.7forawindtunnel

angleofattackof8degrees;forthepresentexperiments,the effectiveangles ofattack

α

are0.2and5.4degrees respec-tively.Additionally,thesuctionpeakclosetotheleadingedgeconfirmsthatblockageeffectsarenegligible[36],asexpected fromthelow ratio ofairfoilthickness tojet width(equal to 0.05).In Fig. 7, theCp distributions obtainedforperforated insertswithl_h rangingfrom1.5to 5mm are alsoplotted. No significant differencesbetweendatameasured forsolid or anyofthepermeableinsertsareobserved;thepreviouslyreported[14,37–39]lossofliftmustbethereforerestrictedtothe contributionproducedbythesolidextentthatisreplacedbypermeablematerial.Forthecurrentexperimentalset-up,that contributionamountstoonly4%oftheoveralllift.

2.6.Acousticmeasurements

Far-fieldacousticmeasurementsareperformedwithaphased microphonearray containing64G.R.A.S.40pHfree-field microphones(frequencyresponse: ± 1 dB;frequencyrange:10Hzto20kHz;max.output:135dBref.20

μ

Pa;nominal

Fig. 7. Pressure coefficient C p along the chord of the airfoil for solid and perforated inserts at a wind tunnel angle of attack of 8 degrees. XFOIL data are

obtained for an angle of attack of 5.4 degrees.

Fig. 8. Microphone distribution within the array (as seen from the back). The airfoil is marked by the grey area. The distance between trailing edge and microphone array plane is 1.01 m. The trailing edge is approximately 30 mm downstream the center microphone. The integration area is shadowed in red. Flow goes from bottom to top. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

phasespreading: ± 3deg.)withintegratedCCPpreamplifiers.Themicrophonedistributionwithintheantenna,shownin

Fig.8,isoptimizedtoreduce boththeMainLobeWidthandtheMaximumSidelobeLevel[40].Thespanwiseand stream-wise diametersofthearray are1mand2mrespectively(Fig.8).Thedistancefromthearray planetotheairfoiltrailing edgedis1.01m.Thecenterofthearrayis30mmbelowthecenteroftheairfoiltrailingedge.

Dataare sampledfor20s ata samplingrateof50kHz. Acousticdataare dividedintimeblocksof5000samples(



t = 0.1s),thus resultingin afrequency resolutionof10 Hz,andwindowed usinga Hanningweighting function with50% overlap.The CrossSpectral Matrixofthemeasured acousticpressure isobtainedbyaveraging spectracomputedforeach sampleblock[41].Therefractionofsoundwaveswithintheshearlayerofthejetiscorrectedusingthemethodproposed bySijtsma[42].Conventionalfrequencydomainbeamforming[43,44]isperformedonasquaregridrangingbetween−2< X/c<2and-2 <Z/c< 2anddistancebetweengridpointsof10mm.Inordertominimizetheeffectofextraneousnoise sources, integrationofthe source mapwithin −0.3<Z/c< 0.3and−0.6<X/c< 0.4(redarea inFig. 8) isperformed. A moredetailedparametric studyontheeffects ofvaryingthespanwise andchordwiseextentofthe integrationregionon theretrievedsoundlevels canbefound inMerino-Martínezetal.[45].Acousticspectraincludedatafrom400Hz,which corresponds to the low-frequency bound for the anechoicity ofthe testing chamber.The frequency high-bound depends onthefree-streamvelocity, andisdeterminedbythepresenceofthecharacteristiclinesourceatthetrailingedgewithin beamforming maps (Fig. 9(a–f)); specifically, data up to 1.4 and 3.8 kHz are reported for measurements at free-stream velocities of 12.5and 34m/s respectively. The aforedescribedacoustic data reduction is oftenemployed intrailing-edge noiseresearch,withsatisfactoryresults[46].Uncertaintyisestimatedin _{± 1}dBbasedonapreviouslyreportedcomparison withsyntheticdata[47].

Beamformingresults are includedin Sections3.1, 3.2 and3.4.Yet, giventhe low-frequencynature (below 400 Hz) of the fundamentaltones described in Section 3.5,the resolution of the array is too low to distinguish trailing-edge noise

Fig. 9. Acoustic source maps measured at U ∞ = 25 m/s for solid and permeable trailing edges described in Section 2.2 . Source maps correspond to a

frequency of 1400 Hz. Integration area is marked by the dashed line. Dynamic range is 14 dB. Permeable inserts cover 20% of the chord length. (a) Solid. (b) l h = 5 mm. (c) l h = 3 mm. (d) l h = 2.5 mm. (e) l h = 2 mm. (f) l h = 1.5 mm.

fromother noise sources.Forthis reason,far-field acousticspectraare obtainedby averaging spectraacquiredby the 64 microphones[48]. Toaccurately resolvetones, the numberof samples ineach sampleblock is increasedto 65536, thus yieldingafrequencyresolutionof0.76Hz.Acousticdatacorruptedbytonesalsopresentinbackgroundnoisemeasurements areremovedfromspectraandreplacedbylinearinterpolationbasedonadjacentpoints.Noiselevelsarenotcorrectedfor soundwavereflections;thereforetheirmagnitudeisnotanalysed.

3. Results

3.1. Scalinglawsfornoisemitigation

InFig.10(a),far-fieldacousticspectraforafree-streamvelocityU∞ =20m/sand

α

=0.2degreesareshown. Dataare presentedintermsofSoundPressureLevelinone-thirdoctavebandsL_{p (1/3)}(referencepressure:p_ref=20

μ

Pa)asafunction ofthechord-basedStrouhalnumberStc =fc/U_∞.Relativelevels



L_{p (}1/ 3)=Lp(1/ 3), solid− Lp(1/ 3), permeablewithrespecttothe

baselineconfigurationareshowninFig.10(b),wherepositivevaluesrefertonoiseabatementwithrespecttothefullysolid airfoil.Similarly toother flow permeablestructures withamorecomplex porearrangement,such asfoams[11],sintered granulates[10]orfelts[5,14],perforatedinsertsyieldlow-frequencynoiseattenuation,withhighernoiseabatementbeing measuredforinsertswithhigherflow permeability.Specifically,thelargestnoise attenuation(equal to11dB)isreported forthemostpermeableinsert(l_{h =}1.5mm)atSt_{c =}6.3.Alsoinlinewithotherstudies[11],acousticspectraforpermeable trailingedgesshowcross-overStrouhalnumbersSt∗_c ,whichsetthehigh-frequencyboundfornoisemitigation.St∗_c depends onthe type ofinsert2 _{Specifically, a higherStc}∗ _{is computedforperforated inserts withhigher permeability,i.e.}

increas-ingthepermeabilityofthetrailingedge notonlyyields higherabatementbutalsoextendsthefrequencyrangeatwhich noisemitigationisobserved.Contrarily,adecreaseinSt_c ∗withincreasingporesize,hencepermeability,hasbeenpreviously reportedforporousedges[9,11](notethatforfoamsthesetwopropertiesareusuallylinked).Thisfindingsuggeststhat dif-ferentmechanismsareresponsibleforexcessnoiseinporousandperforatededges.Thisaspectwillbefurtherinvestigated laterinthissection.

InFig.11(a)and(b),far-fieldacousticspectraforafree-streamvelocity U_∞ =20m/sand

α

= 5.4degreesareshown. Similarlytodataforlowerliftconditions(Fig.10(a)and(b)),perforatedtrailingedgeswithhigherpermeabilityyieldhigher noisemitigation belowSt∗_c .Maindifferenceswithrespecttotheprevious configurationconcerna decreaseofthehighest

2 Note that for the design with l

)

b

(

)

a

(

Fig. 10. Far-field acoustic spectra for the perforated inserts (with s / c = 0.2) at U ∞ = 20 m/s and α= 0.2 degrees. (a) Absolute levels. (b) Relative levels

with respect to the baseline configuration Lp(1/3) = L p(1/3),solid − L p(1/3),permeable .

)

b

(

)

a

(

Fig. 11. Far-field acoustic spectra for the perforated inserts (with s / c = 0.2) at U ∞ = 20 m/s and α= 5.4 degrees. (a) Absolute levels. (b) Relative levels

with respect to the baseline configuration Lp(1/3) = L p(1/3),solid − L p(1/3),permeable .

noiseabatement(forinstance,themostpermeableinsertnowyields



Lp,(1/3) =8dBatStc =12.5)andanoverallincrease

ofSt∗c.

InFig.12(a),narrow-bandnoiseattenuation



L_p forU_∞ of12.5,15,20,25,30and34m/sareplottedasa functionof f.Onlydatacorresponding toinsertswithlh =2,3and5mmareshownforthesakeofclarity.Forthesamereason, only

oneeverytenpointsaredepicted(



f=100 Hz).AsseeninFig.12(b),



Lp datameasuredfordifferentfree-stream veloc-itiescollapsewhenplottedasafunction ofStc ;for



Lp ,therelationshipbetweenfrequencyandU∞ isthuslinear3_.After

collapsing,dataforinsertswithdifferentflowpermeabilitiesyieldcurvesthat definethenoiseattenuationperformanceof each design independently ofU_∞. Inline withresults presentedabove, characteristiccurves forinserts withhigher per-meabilityallowforhighermaximumnoiseattenuationandSt_c ∗.Thisfindingallowstopredictthefrequencyformaximum noiseattenuation,orthefrequencyrangefornoisereductionfordifferentvelocities.Althoughnotshownhereforthesake ofbrevity,thesamemethodologyyieldsasatisfactorycollapseofdatameasuredat

α

=5.4degrees.Forcompleteness,these characteristiccurveshavebeencomparedtonoise mitigationdatapredictedbythelower errorsymbolicregression mod-elsdescribedinSarradjandGeyer[49].However,theoverallagreementisnotsatisfactory;thereisnocollapseofpredicted noisemitigationdatawithStc ,andhighernoisemitigationmaximaarepredictedformaterialswithdecreasingpermeability, contrarilytoexperimentaldatapresentedinthismanuscript.

3 For the current analysis, the chord is used as length scale for convenience. An attempt to further collapse data for different inserts using length scales

) b ( ) a (

Fig. 12. Collapse of noise attenuation Lp = L p,solid − L p,permeable for free-stream speeds ranging between 12.5 and 34 m/s and α= 0.2 degrees for inserts

with l h = 2, 3 and 5 mm and s / c = 0.2. (a) Lp as a function of f . (b) Lp as a function of St c .

Fig. 13. Sketch depicting collapsed Lp = L p,solid − L p,permeable data for the insert with l h = 2.5 mm at α= 0.2 degrees and free-stream speeds ranging from

12.5 to 34 m/s. The B-Spline fitted to experimental data is shown as a thick dashed line. In the sketch, low and high-bound cross-over Strouhal numbers St 

c and St ∗c , the Strouhal number range for noise mitigationSt ◦= St ∗c -St



c , the maximum noise attenuation Lp,max and its chord-based Strouhal number

St +

c are also depicted. The thin dotted line indicates extrapolated data.

Thecharacteristiccurvesshownaboveallowtostudythechangeofrelevantnoisemitigationpropertieswiththetrailing edge permeability. Specifically, low St_c andhigh-bound St∗_c cross-over Strouhal numbers, the Strouhal number range for noisemitigation



St◦c = St∗c-Stc,themaximumnoisemitigation



Lp ,max andtheStrouhalnumberatwhichoccursSt+c ,as

definedin Fig.13, are analysed inthe following.To compute theseproperties,cubic B-splines [50] with8 knots andC2 continuity[51]are fittedto experimentaldata.InthetestcaseswhereSt_c orSt∗_c are notpresentwithinthemeasured Stc

range,thesplineisextendedwithconstantslope,asshowninFig.13(thindottedline).

Firstly,thechangeinmaximumnoiseattenuation



Lp ,max withpermeabilityK,showninFig.14(a),isanalysed.



Lp ,max

dataformetalfoamedges,whichalsocollapsewithSt_c arealsopresentedforcomparison.Tofacilitatetheinterpretationof results,



Lp ,maxdataarefitted4withahyperbolictangentfunction,asdefinedinEq.(5)



Lp, max=

γ

1tanh

(

γ

2K

)

(5)

where

γ

1 definestheasymptotic



Lp ,max value and

γ

2 theslope ofthefunction withinthe low permeabilityrange.The

reasonstochoosethisfunctionare twofold:firstly,itadequatelymimicstheasymptoticcharacterofexperimentaldatafor increasingpermeabilityoftheedges;secondly,itpassesthroughthepoint K=0m2_,

_

_L

p ,max =0dB,properlydescribing

baseline data.These curvesyield valuable information on thetype ofmicrostructure that maximizes noise mitigation.It isobserved that, forperforatedinserts at

α

= 0.2 degrees,increasing the permeabilityto 3.5 × 10−9 _m2 _{causes up to}

(a) (b)

Fig. 14. Change in maximum noise mitigation with permeability for perforated and metal foam inserts at α= 0.2 and 5.4 degrees. (a) Lp,max . (b) St +c .

8.3dB noisemitigation. Furtherincreasing thepermeabilityoftheinsert bringsfew additionalnoise attenuationbenefits (approximately 1dBmore, upto anasymptoticvalue of9.3dB atK =6 × 10−9 _m2_{). Furthermore,increasing}

_α

_{to 5.4}

degreespenalizesnoisemitigation:



L_p ,maxvaluesmeasuredatahigher-liftconditionarelowerthanthosecomputedat0.2

degreesindependentlyofthepermeability.Interestingly,lower-permeabilityvariantsarelesssensitivetoachangein

α

:for instance,fortheleastpermeableperforatedinsert (l_h = 5mm)increasing

α

yieldsa



Lp ,max lossequalto 0.5dB,whilst

forthemostpermeableinsert(l_{h =}1.5mm)thedecreaseisequalto2dB.Giventheliftdecreaseanddragincreasewhen employinginsertswithhigherpermeability[5,38],aswell astheir highersensitivitytochanges in

α

,theemploymentof perforatedinsertswithmoderateflowpermeability(belowathresholdofK=3.5 × 10−9_m2_{)isrecommended.Forinserts}

manufacturedwithmetalfoams,similarfeaturesarefound:maximumnoiseattenuationincreasesrapidlywithpermeability up toapproximately8.4dB (fora Kthresholdof1 × 10−9 _m2_{,lessthanone third ofthevalue reportedforperforated}

pores);further increasing Konly yields a1dB gain in



Lp ,max,up toa maximumnoise mitigationof9.4 dB.Bothmetal

foamandperforatedtrailingedges shareahighboundfor



Lp ,maxof9.3–9.4dB;yet,foragivenK(below6 × 10−9 m2)

an additional increase ofup to 4 dB (for edges withK between0.5 and2 _{× 10}−9 m2_{) can be achievedby employing}

a random porearrangement. Similarly, to obtainsimilar



Lp ,max levels, perforated insertsmust be more permeablethan

metal foamtrailing edges.In view ofthesefindings, thepermeabilityofthe trailingedgecan be thereforeconsidered as agoodindicator fornoisemitigation effectiveness,butitisinsufficienttofullycharacterizethenoisescatteringoftrailing edgeswithsignificantmicrostructuraldifferences.Otherparametersdescribingthedifferentporeorganizationofpermeable materials,suchastortuosity,needtobeconsideredtofullycharacterizetheirnoisescattering.Thisanalysiswillbepresented inSection3.2.

InFig.14(b),thechord-basedStrouhalnumberatwhich



Lp ,max occurs,St+_c ,isshownforperforatedinsertsat

α

=0.2

and5.4degrees,andmetalfoaminsertsat

α

=0.2degreesasafunctionofK.Forperforatedinsertsat

α

=0.2degrees,an approximatelylineardecreaseofSt+_c withKismeasured:theleastpermeableinsertproducesmaximumnoiseattenuationat St+_c =11,whilethemostpermeableinsertyieldsSt+_c =7.5.Therefore,perforatedtrailingedgeswithincreasingpermeability notonlyyieldhigher



Lp ,maxbutalsomitigatelow-frequencynoisemoreefficiently.Increasingtheangleofattackto

α

=5.4

degreesleadstoaconstant St+_c value of11forall K; similarlyto



Lp ,max,insertswithhigherflowpermeabilityare more

sensitive toachangein

α

.Regarding metalfoams, low-permeabilityinserts(K= 0.5 × 10−9 _m2_{) yieldmaximumnoise}

attenuationatSt+_c =9.5,closetothevalueforperforatededges.However,increasingKto1.5 × 10−9_m2_{rapidlybringsSt}+ c

to5.7;forhigherflowpermeabilitySt+_c isapproximatelyconstant. Therefore,thechoice ofan adequateporearrangement allows not only to obtain different maximum noise attenuation levels (below a certain K threshold), but also to target specificfrequencies.

InFig.15,thelower-boundcross-overStrouhalnumbersSt_c ,computedforperforatedandmetalfoaminsertsat

α

=0.2 and5.4degrees, areplottedasafunction ofK.Forperforatedinserts at

α

₌0.2degrees, St_c increasesfrom0.04to 2.44 withinthemeasuredpermeabilityrange.Increasingtheangleofattack,oremployinginsertswithrandomporedistribution (i.e. metal foams) yieldhigher St_c ; specifically, St_c values between 0.96 and2.68 are measured for the former case,and between2.28and2.96forthelatter.Itcanbethereforeconcludedthattoguaranteenoise mitigationwithintheverylow frequencyrangeonemustemployedgeswithlowpermeabilityandorderedporearrangement.

In Fig. 16(a), high-bound cross-over Strouhal number Stc∗ data are presented. For this quantity, a linear-like increase

Fig. 15. Change in St c with permeability for perforated and metal foam inserts at α= 0.2 and 5.4 degrees.

(a) (b)

Fig. 16. (a) Change in St ∗

c with permeability for perforated and metal foam inserts at α= 0.2 and 5.4 degrees. (b) Sketch depicting typical roughness

noise features for porous edges. Black line represents baseline (smooth surface) trailing-edge noise. Dashed lines illustrate roughness noise contributions for increasing pore size d c , i.e. higher roughness.

withhigher permeabilityalso mitigatenoise up tohigher frequencies (in line withresults shownin Fig. 10(a) and(b)). Additionally,increasingtheangleofattack to

α

=5.4degreesfurtherincrementsSt∗c.Formetalfoams,adifferenttrendis

reported:alowerSt∗_c ismeasuredforincreasingpermeabilityoftheinsert.TheoppositetrendsforthevariationofSt∗_c with Kindicatethatdifferentmechanismsareresponsibleforhigh-frequencynoiseinperforatedandmetalfoamedges.Formetal foamedges,high-frequencyexcessnoise iscausedby surfaceroughness[9,11];ithasbeenpreviouslyshownthata larger poresize,thatoftentranslatesinlargerK,generateshighernoiseandbecomesmorerelevantatlowerfrequencies[52],in agreementwithresultsinFig.16(b).Itisalsoimportanttoremarkthat,forthepresentdata-set,narrowbandcontributions areonlyfound forthemostpermeableperforatedinsert;however,the low-frequencynatureofthesetonessuggeststhat theyarenotdrivenbyroughness.TheirnaturewillbeaddressedindetailinSection3.5.

TheresultingSt_c rangefornoiseabatement



St◦₌St_c ∗-St_{c ,}measuredforperforatedandmetalfoaminsertsatanglesof attackof0.2and5.4degrees,isshowninFig.17.ResultsaresimilartothosefoundforSt∗_c ,i.e.,perforatededgeswithhigher permeabilityyieldacousticbenefitswithinawiderfrequencyrange,andincreasing

α

furtherextendsit(approximatelyan increaseof



St◦ ₌4with

α

ismeasuredindependentlyofK).Formetalfoamedges,adecreasein



St◦ withKisobserved instead;hence,contrarily toperforatededges, morepermeable metal foamedges yield noisemitigation within a smaller frequencyrange.ThesimilaritybetweenFig.16(a)andFig.17suggeststhat



St◦ isparticularlysensitivetotheappearance ofnoisegenerationmechanismsotherthanedgenoise,thatgeneratehigh-frequencyexcessnoise.

Fig. 17. Change in St ◦

c with permeability for perforated and metal foam inserts at α= 0.2 and 5.4 degrees.

Fig. 18. Maximum noise attenuation Lp,max as a function of K and τY for α= 0.2 deg. 3.2. Effectoftortuosityonnoisemitigation

FollowingthediscussioninSection3.1,tortuosity

τ

Yisproposed,inadditiontoK,asanadditionalindicatorforthenoise

mitigationcapabilitiesofpermeabletrailingedges.InFig.18,



Lp,maxdataareplottedasafunctionofflowpermeabilityand

tortuosityforperforatedandmetalfoamtrailingedges.

Itisobservedthat,foragivenpermeability,micro-structureswithmoretortuousporepaths,i.e.foams,gainupto4dB noiseattenuationwithrespecttostraightchannelledinserts.Tothisend,highlytortuousporearrangements,suchasthose derived fromsintering[53] orcastingaround space-holders[54]manufacturingprocesses,mightyield additionalbenefits forlow-noiseapplications.Forthecurrentexperimentalset-up,atrailingedgewithpermeabilityofK=1 × 10−9_m2 _and

tortuosity

τ

_{Y ࣃ} 1.15isagoodtrade-off betweenaerodynamicandbroadbandacousticperformance. 3.3. Scalingoffar-fieldacousticpressurewithfree-streamvelocity

The exponentmforthe scalingoffar-fieldmean-squared acousticpressurewithfree-stream velocityis nowanalysed. ThisquantityiscomputedfittingtheOverallSoundPressureLevel(OSPL)[55],calculatedas

OSPL=10log₁₀ f c

10L p(1/3)/ 10dB ₍₆₎

toexperimentaldatameasuredatdifferentspeeds.Toisolatebroadbandnoisemitigation,onlyfrequencybandswherenoise abatementwithrespecttothebaselineconfigurationismeasuredareconsidered.Forthesamereason,themostpermeable perforated(l_h =1.5mm)andmetalfoam(dc =1200μm)insertsareexcludedfromtheanalysisbecause,aswillbeseenin

Fig. 19. Analysis of the exponent m of the scaling low of the far-field acoustic pressure with free-stream velocity U ∞ . Change in m with permeability K for

metal foam and perforated inserts at α= 0.2 degrees.

Section3.5,acousticspectrafortheformercontaintones;forthelatter,alow-frequencybroadbandhumpismeasured.The fitofexperimentalOSPLdatatoU_∞isperformedemploying

OSPL=m10log10



_U ∞ Uref

+

η

(7)

with

η

beinganadditional fitcoefficient andUref=1m/sforconvenience. Forthebaseline insert,thefit yieldsm=5.1,

inlinewiththeoretical[56,57]andexperimentalresults[58].Thecomputedexponentmisplotted(togetherwiththe un-certainty,estimatedasthe95% confidenceboundsofthe fit) asafunction ofK inFig.19 forperforatedandmetal foam insertsat

α

=0.2degrees.Inspiteofthemanyspectraldifferencesdescribedabove,exponentsmformicro-structureswith similarflowpermeabilityarecomparable(withintheir uncertaintyrange)independentlyoftheporearrangement.This re-sultsuggeststhatalthough specific noisemitigation features (St_{c ,}St_c ∗_,



L_p ,max,etc.)forpermeableinserts relyoncertain

micro-structural properties,they all act upon noise generationthrough the samemechanism, i.e. decreasingthe acoustic impedancejump at the surface discontinuity andcreating distributed noise scattering [9]. Data for both perforated and metalfoamedgesincreaselinearlyfromm₌5_.1(atK₌0m2_{,solidcase)tom=5.9(atK=3× 10}−9_m2_{).Consequently,m}

valuesarefitted5_{toaline,definedasfollows}

m=

χ

1 K+

χ

2 (8)

Thelinearincreasesuggestsintrinsicchangesinthenoisesourcenature(fromadipoleoveranon-compacttoacompact surface),andmightbeexplainedbyamoreeffectivedistributionofnoisesourcesalongthepermeableextentforincreasing K,thatdecreasestheratioofeffectivechordlengthtonoisesourceextension,i.e.,itpromotesthecompactnessoftheairfoil. 3.4.Effectoftherelativepermeablechordlengthonnoiseabatement

Testsarecarried outto assesstheeffectofachangeinthe permeablechordwiseextentsonnoise mitigation.Tothis aim,besidesdatagatheredfors/c=0.2(40mm),additionalexperimentsarecarriedoutforperforatedinsertswithl_h =2 and5mm;specifically,theseinsertsaretapedatbothsides,leavingonlys/c₌0.1(20mm)ands/c₌0.05(10mm)exposed totheflow. Asseen inFigs.20(a)and(b),noisemitigation dataforvaryingU_∞ alsocollapsewhennon-dimensionalizing thefrequencyasStrouhalnumber,yielding thecurvesalreadydescribedinSection3.1.Similarly,relevantparameterssuch asSt_{c ,} St∗_{c ,}



St◦,



L_p ,max and St+c are computed fitting experimental datato a spline. For data corresponding to shorter

permeablelengths,thelower-boundcross-overStrouhalnumberSt_c ,isdirectlycomputedfrommeasureddata,i.e.,nodata extrapolationisrequired.

ResultsareshowninFig.21(a)and(b),whereSt_{c ,}St∗_{c ,}



L_p ,maxandSt+c areplottedasafunctionofthepermeablelength

tochord ratios/cforperforatedinsertswithl_h = 5and2mm respectively.Fortheperforatedinsertwithl_h =5 mm,an overalldecreaseinallSt-basedparametersforincreasings/cisfound(Fig.21(a));hence,alongerpermeableinsertwiththis porearrangementdoesnotbroadenthefrequencyrangefornoiseabatement,butisbeneficialtotacklelower frequencies. Datafortheperforatedinsertwithlh =2mmshowadecreaseofStc withs/c;yet,St∗c hasanapproximatelyconstantvalue

of25foralls/c.Consequently,forthisinsertthenoisemitigationrangeisextendedforincreasings/c.

ThesefindingsareanalysedmoreindetailinFig.22(a),wherethefrequencyrangefornoisemitigation



St◦_c isplottedas afunctionofs/c.Fortheperforatedinsertwithl_{h =}5mm,increasings/cyieldsfewadditionalbenefitsin



St◦_c ;fortheone

(a) (b)

Fig. 20. Collapse of Lp = L p,solid − L p,permeable with St c for perforated inserts with l h = 5 mm and permeable extent s / c = 0.2 and s / c = 0.05. Data are

measured at U ∞ ranging from 12.5 to 34 m/s and α= 0.2 degrees. Dashed lines represent the B-Spline fitted to experimental data. Thick dashed lines

indicate interpolated data. Thin dashed lines indicate extrapolated values.

Fig. 21. Change in maximum noise attenuation Lp,max with permeable chordwise length s / c and chord-based Strouhal number St c for perforated inserts

with l h = 5 (a) and 2 mm (b) at α= 0.2 degrees.

withl_h =2mm,aslightextension ofthefrequencyrange(from23ats/c=0.05to24.5ats/c=0.2)isreportedinstead. Thisresultevidencesthatincreasing s/cisnotaconvenientwaytoachievenoisemitigationthroughoutawiderfrequency range(tothisend,increasingKismoreeffective),butrathertoenhanceabatementuponlowerfrequencies.

Finally,maximumnoiseabatement



Lp ,max dataareplottedinFig.22(b).AsinSection3.1,experimentaldataarefitted6

with



Lp, max=

ξ

1tanh



ξ

2 s c

(9)

(a) (b)

Fig. 22. Change in the frequency range for noise mitigation St ◦

c (a) and maximum noise attenuation Lp,max (b) with permeable chordwise length s / c for

perforated inserts with l h = 2 and 5 mm ( α= 0.2 deg.).

Resultsshow thatwiths/c =0.05, insertswithl_h =2 mmand5mmabateup to1.6and6.7dB respectively.Further increasing s/c yields negligible additional abatement(lower than 1dB). Sincea longer permeable extentwould decrease liftand increase drag, this value represents a goodtrade-off between noise mitigation andaerodynamicperformance. It isinterestingto notethat forthe s/c= 0.05configurationthepermeableextentapproximatelycoincides withthe bound-arylayerthicknessatthetrailingedgeforthebaseline configuration(previousstudies[9,12]report avalue of9.3mmat U_∞=20m/s);whetherthisobservationcanbegeneralizedorismerelyacoincidenceforthecurrentexperimentalset-up willbe addressedin futureresearch. It is alsointeresting to note that some ofthe features derived fromdecreasing the lengthofthepermeableinsert,such asthedecreasein



Lp ,max,ortheincreaseinSt∗c , areinlinewiththosereportedin

Section3.1forincreasing

α

.Thismightberelatedtothepassageofsteadyflowthroughpermeablemediaduetoapressure imbalancebetweenpressure andsuction side, illustrated,for instance,by MößnerandRadespiel [59],that decreases the effectivelengthofthepermeableextent.

3.5.Appearanceofshedding-relatedtones

Inspiteofthejustreportedbenefitsregardingbroadbandnoisemitigation,forpermeableinsertswithperiodic arrange-ment of the channels andhigh flow permeability, an undesired toneis measured independently of the angle ofattack. Specifically, such afeature is found infar-fieldacoustic spectra forthemost permeableperforatedinsert (l_h = 1.5mm). InFig.23,narrowbandacousticmeasurements forafree-streamvelocity of25m/sandanangleofattack of0.2degrees arepresented,togetherwithbackgroundnoise measurementsanddataforthel_h = 5mmandreferenceconfigurations.A strong(+30dBoverbroadbandnoiselevels)tonalcomponentismeasuredatafrequencyof208Hz,withharmonicsat416 and624Hz.

InFig.24(a), narrow-bandspectra forthelh = 1.5mm insertatU∞ = 15,20,25,30 and34m/sare shown. Onlyat

U_∞=15m/snotonesareobservedwithinthemeasuredfrequencyrange.Whenfisscaledemployingthethicknessofthe airfoilatthesolid-permeablejunctionh=15.7mmandthefree-streamvelocity,alltonescollapse(Fig.24(b)).Theanalysis yieldsafundamentaltoneatSth,1=0.13,withvisibleharmonicsuptoSth,i=0.52.Suchvaluesareinlinewiththose

typi-callyfoundforblunttrailing-edgenoise[14,58],inwhichasimilarpeakarisesduetosheddingfromtheblunttrailingedge. Therefore,thenature ofthistoneseemstoberelatedtotheappearanceofperiodicflowphenomena, i.e.vortexshedding fromtheapparent blunt endofthe airfoilaluminiumbody,duetothe highpermeabilityandperiodicpore arrangement oftheinsert.Theintroductionofacertaindegreeofrandomnessintheporedistributionmightaltertheorganizationand periodicityofvortices,thuspreventingtheappearanceofstrongtones.Inagreementwiththishypothesis,themetal foam withd_{c =}1200 μmandtheperforatedinsertwithl_{h =}1.5mmhavecomparableflowpermeability(K_ࣃ 6 _{× 10}−9m2_);

yet,tonesaresolelyreportedforthelatterinsert.

Thisaspectisfurtheraddressedinanadditionaltestinwhichthepermeableinsertistaped,leavingexposedtotheflow apermeablechordwiseextent equaltos/c₌ 0.1₌ 20mm.Asimilar analysisforthistestcase(Fig. 25(a)and(b))(with athicknesshof9.5mm)revealsastrongpeakatSth,1= 0.10,andsubsequentharmonics.Notethatforthisconfiguration,

additionalweakertonesappeararoundthefundamentalone.Theoriginofthesemightbeinthetapingprocedure:iftape isnot perfectlyparalleltothetrailingedge,orisapplieddifferentlyatsuctionandpressuresides,onemightexpectslight

Fig. 23. Narrow band far-field acoustic spectra for the solid and perforated inserts with l h = 5 mm and l h = 1.5 mm at U ∞ = 25 m/s and α= 0.2 deg.

(a) (b)

Fig. 24. Collapse of tonal noise with St h for the perforated insert with l h = 1.5 mm at free-stream velocities ranging from 15 to 34 m/s and α= 0.2 deg.

Permeable surface extends for the last 20% of the chord (40 mm). (a) Far-field acoustic spectra as a function of f . (b) Far-field acoustic spectra as a function of St h .

variations inthe solid-permeable junction thickness along the span; hence,some deviations fromthe fundamentaltone frequency.

Finally,inFig.26,bothSt_h ,1valuesareshownasafunctionofthethicknessratioh/

δ

∗.Displacementthicknessdataare

obtainedfromaprevious boundarylayercharacterization [12],performedat20m/semployinghigh-speedParticle Image Velocimetry;onlydatameasuredatthisfree-streamvelocityareshown. Experimentaldataarecomparedtotheempirical relationforSt_h ,1proposedbyBrooksetal.[58],definedas

Sth, 1=

0.212− 0.0045



1+0.235

(

h/

δ

∗

₎

−1_{− 0.0132}

₍

_h/

δ

∗

₎

−2 (10)

where



isthesolidangle(in degrees)betweentheslopingsurfacesupstreamofthetrailingedge.Forpermeableextents s/c₌0.1ands/c₌0.2thesolidangles



arerespectively21and19degrees.Forcomparison,datareportedintheoriginal studyfor a flatplate (



= 0 deg.) anda NACA 0012 airfoil(



= 14deg.) are also included.A fairagreement between measureddataandpredictionsisfoundindependentlyofthepermeableextent;theshedding-relatednatureofthetoneis thereforeconfirmed.

Fig. 25. Collapse of tonal noise with St h for the perforated insert with l h = 1.5 mm at free-stream velocities ranging from 15 to 34 m/s and α= 0.2 deg.

Permeable surface extends for the last 10% of the chord (20 mm). (a) Far-field acoustic spectra as a function of f . (b) Far-field acoustic spectra as a function of St h .

Fig. 26. Trailing-edge thickness based Strouhal number of the first peak St h,1 as a function of thickness ratio h / δ∗. The empirical prediction proposed in

Brooks et al. [58] ( Eq. (10) ) is plotted for a flat plate ( = 0 deg.), a NACA 0012 airfoil ( = 14 deg.) and a NACA 0018 airfoil with s / c = 0.1 ( = 21 deg.) and s / c = 0.2 ( = 19 deg.). Reported St h,1 values [58] for a flat plate and a NACA 0012 airfoil are also depicted for comparison.

4. Conclusions

In thismanuscript, acoustic measurements on a NACA 0018 airfoil(with chord c = 0.2 m) equipped with solid and permeabletrailing-edgeinsertsareperformedbymeansofamicrophoneantenna.Toobtaincriteriaformaximizingnoise abatementwhilelimitingchangesinaerodynamicperformance,trailingedgeswithstraightopenchannelsthatlinksuction andpressuresidesaremanufactured.Forcomparison,trailing-edgeinsertsmanufacturedwithopen-cellmetalfoams,with comparableflowpermeabilitybutrandomporedistribution,arealsoanalysed.

Far-field noise spectra reveal the collapse of



Lp , defined as the difference between noise scattering from solid and permeableedges,whennondimensionalizing frequencywithchordandfree-stream velocity.Collapsed dataproduce sine-like



L_p curves that define the acoustic benefits of each trailing edge independently of flow speed. From thisanalysis, it stems that the change inmaximum noise mitigation



Lp ,max withpermeability K can be accurately describedwith a

hyperbolictangent(



Lp, max=

γ1

tanh

(

γ2

K

)

,where

γ

1and

γ

2 arefittingcoefficients),i.e.significantnoisemitigationcan

be achieved with moderate K, but it saturates above certain K thresholds, that depend on the type of edge employed. Specifically,permeabilitythresholdsof3.5 × 10−9 _{and1 × 10}−9_m2 _{arerespectivelyobtainedforperforatedandmetal}

foaminserts.Theemploymentofedgeswithpermeabilityequaltothesethresholdsisthereforerecommendedtominimize aerodynamicpenalty(i.e.,decreaseinliftandincreaseindrag)duetotheuseofpermeablematerialsinliftingdevices,to

maximize thefrequencyrangefornoisemitigation,andtoprevent anexcessive downgradeoftheacousticbenefitswhen increasingtheangleofattack.Furthermore,thetortuosityofthetrailingedgeisproposedtoaccountfordifferencesinnoise mitigation reportedforedgeswithsimilarpermeabilitybutdifferentporearrangement.Thismanuscriptalsostressesthat increasingthechordwiseextentofthepermeableregionsbeyonds/c₌0.05(whichinthecurrentstudyisinlinewiththe boundarylayerthicknessatthetrailingedgeforthebaseline configuration)doesnotextendthefrequencyrangefornoise mitigation,norincreasesitsmagnitudesignificantly.Finally,itisalsoconfirmedthatinsertswithhighpermeabilityandan orderedporearrangementproduceextremely loudtones,whicharecausedbyvortex-sheddingfromthebluntedgeofthe airfoilsolidbody.

Takentogether,thesefindingssuggest thatapermeabletrailingedgeinsertwithpermeabilityequalto1 × 10−9 _m2_,

withatortuosityof1.15(whichisinturnbeneficial toavoidtones) andwiths/c= 0.05yieldsthebesttrade-off between broadbandnoisemitigationandaerodynamicloss.Futureworkwilladdressoptimaltrailing-edgepropertiesformore real-isticconditions,i.e.,camberedairfoilswithlongerchord,aswellasthedefinitionofthemicro-structuralandfluidfeatures necessarytopreventsheddingfrompermeableedges.

DeclarationofCompetingInterest

The authors declare that they have noknown competing financial interests orpersonal relationshipsthat could have appearedtoinfluencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

AlejandroRubio Carpio: Conceptualization, Methodology, Data curation, Writing - original draft. Francesco Avallone: Conceptualization, Supervision,Methodology, Writing - review & editing.DanieleRagni: Conceptualization, Methodology, Supervision,Writing - review & editing.MirjamSnellen: Supervision,Methodology, Writing - review & editing.Sybrand vanderZwaag:Supervision,Methodology,Writing-review&editing.

Acknowledgments

TheauthorswouldliketoacknowledgeLourencoLimaPereiraforhisassistanceduringthecharacterizationoftheinflow conditionsandFrankSchilderforhisadviceandfeedbackforthemanufacturingoftheinserts.

AppendixA. AdditionalScalingLaws

InSection3.1,thecollapseofnoisemitigationdatafordifferentfree-streamvelocitieswhenplottedasafunctionofStc

isshown.Intheanalysis,thechordoftheairfoilisemployedaslengthscaleforthesakeofsimplicity.However,anattempt tocollapsedatafordifferentpermeableinsertsemployinglengthscales basedonhydraulicproperties,suchas√K_,1/Cor F=KC wasalsoperformed.

Resultsare showninFigs.A.1(a),(b)and(c),wherecross-overStrouhal numbersbasedonthesequantitiesSt∗√

K , St∗1/C

andSt∗_F arerespectivelydepictedasafunctionofthecorrespondinghydraulicpropertyforperforatedandmetalfoaminserts at

α

=0.2and5.4degrees.Asseenintheseplots,employinghydraulicpropertiesdoesnotremovethedependencyofSt∗ onthesequantitiesneitherforperforatednormetalfoaminserts.Yet,forallthreepropertiestherelationseemstobelinear. Interestingly,forthecollapsewith1/Caslengthscale,alldataseemtocollapseinalineindependentlyofholearrangement orliftingcondition.

SimilarphenomenaarefoundfortheanalysisoftheStrouhalnumberformaximumnoiseattenuationSt+.InFigs.A.2(a), (b)and(c),St+√

K ,St+1/C andStF +arerespectivelyplottedasafunctionofhydraulicproperties.

(a) (b) (c)

Fig. A.1. Scaling of St ∗

c with length scales derived from the permeability characterization. (a)

√

(a) (b) (c)

Fig. A.2. Scaling of St + with length scales derived from the permeability characterization. (a) √ _{K . (b) 1/ C . (c) F = KC.}

AppendixB. FittingCoefficients

Forcompleteness, the coefficients employed for the fit of experimental data to Eqs. (5), (8) and (9) are respectively presentedinTablesB.1,B.2andB.3,togetherwiththecoefficientofdeterminationR2_{,thatgivesameasureofthegoodness}

ofthefit.

Table B.1

Coefficients ( γ1 , γ2 ) (with 95% confidence bounds) and correlation coefficient

R2 for the fit of experimental _L

p,max and K data to Eq. (5) .

Case γ1 (dB) γ2 ×10 −9 (m −2 ) R 2

Perforated, α= 0.2 deg. 9.311 ± 0.984 0.388 ± 0.084 0.995 Perforated, α= 5.4 deg. 7.310 ± 0.699 0.456 ± 0.098 0.988 Metal Foam, α= 0.2 deg 9.358 ± 1.221 1.484 ± 0.779 0.943

Table B.2

Coefficients ( χ1 , χ2 ) (with 95% confidence

bounds) and correlation coefficient R 2 for the fit

of experimental m and K data to Eq. (8) . χ1 ×10 −9 (m −2 ) χ2 (-) R 2

23.306 ± 7.535 5.215 ± 0.142 0.8575

Table B.3

Coefficients ( ξ1 , ξ2 ) (with 95% confidence bounds) and correlation coefficient R 2 for the

fit of experimental Lp,max and s / c data to Eq. (9) .

Case ξ1 (dB) ξ2 (-) R 2

lh = 5 mm ( K = 5 × 10 −10 m 2 ) 2.117 ± 0.964 19.834 ± 32.040 0.945

lh = 2 mm ( K = 31 × 10 −10 m 2 ) 7.553 ± 2.513 27.923 ± 50.391 0.869

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