Numerical predictions of low Reynolds number compressible aerodynamics

an author's

https://oatao.univ-toulouse.fr/27054

https://doi.org/10.1016/j.ast.2019.05.064

Desert, Thibault and Jardin, Thierry and Bézard, Hervé and Moschetta, Jean-Marc Numerical predictions of low

Reynolds number compressible aerodynamics. (2019) Aerospace Science and Technology, 92. 211-223. ISSN

1270-9638

Numerical

predictions

of

low

Reynolds

number

compressible

aerodynamics

T. Désert

a

,

b

,

T. Jardin

b

,

∗

,

H. Bézard

a

,

J.M. Moschetta

b

a_{ONERA,Toulouse,France}

b_{ISAE-SUPAERO,UniversitédeToulouse,France}

a

b

s

t

r

a

c

t

Keywords:

LowReynoldsnumber Compressible Lowpressure Airfoils Rotors Mars

Interest inlow Reynoldsnumber compressible flows is emerging duetoprospective applications like flightonMarsandinthestratosphere. However,verylittleknowledgeisavailable,bothregardingthe flowphysicsunderlyingthisuniqueregimeandtheaccuracyofnumericalmethodsforitsprediction.In thispaper, lowandhighfidelitynumericalapproachesare comparedwithexperimentalmeasurements onbothairfoilsandrotorsinthelowReynoldsnumbercompressibleflowregime.Itisshownthatlow fidelityapproachesaresuitedtoaerodynamicoptimizationdespitehighviscousandcompressibleeffects. Inaddition,highfidelityapproacheshelprevealuniqueflowfeaturesofthisregime.

1. Introduction

The exploration of the planet Marswith Rovers startedmore than a decade ago, allowing a more detailed description of the planetsurface than what was possibleso far withMarsorbiters. However,amajordrawbackoftheseRoversistheirrelatively low speed(partlyduetoharshterrain)which,untilnow,onlyallowed themto explorea few tens ofkilometers- tobe compared with the 21,000 km planet circumference. One way to facilitate sur-faceexplorationwouldbetouseautonomousflyingvehicleswhich couldact asscoutsfortheRovers. Unfortunately,the atmosphere ofMarsisfarfromflight-friendly.Ontheonehand,itsdensityis lowwhich(1)resultsinlowReynoldsnumberflowsthatpromote bothflowseparationandhighviscous dragand(2)requirestobe compensatedforbyhigherspeedoftheliftingsurfacetoprovidea sufficientliftforce.Ontheotherhand,itscompositionand temper-aturearesuch thatthespeed ofsoundisrelativelylow,whichis conducivetosupersonicflowregimes.Asaconsequence,atypical flyingvehicledesignedforMarsexplorationwouldoperateinthe low Reynolds number, compressible flow regime, for which very littleknowledgeisavailable.

Table 1 shows properties of the atmosphere of Mars at the groundlevelandcomparesitwithpropertiesoftheatmosphereon Earthatsealevel and30kmabovesealevel(stratosphere). Note

*

Correspondingauthor.

E-mailaddress:thierry.jardin@isae.fr(T. Jardin).

thatthepurposehereistoshowordersofmagnituderatherthan precisetime-averagedvaluesofhighly fluctuatingproperties.It is shownthatthepropertiesat30kmabovesealevelarequite sim-ilartothoseonMars,atgroundlevel,suggestingthatvehicles fly-inginthestratospherewouldoperateundersimilar conditionsto thosedescribedabove,i.e.inthelowReynoldsnumber, compress-ibleflowregime,albeitwithhighergravityforce.Thedevelopment ofstratosphericvehicleshasalsorecentlygainedinterestfor appli-cationslikeEarthobservation,telecommunicationsandnavigation. Overall, low Reynoldsnumber, compressible flow regimes are verypoorlydocumentedbecauseoftherelativelynewapplications towhichtheyarerelated.ApartfromMarsandstratosphericflight, the development of highspeed trainsin low pressure tube (e.g. Hyperloop) couldgreatly benefitfromdeeper knowledgeofthese regimes,aswell asliquidatomizationwheremicro-sizeddroplets are formed and travel athigh speed. Yet, a few authors brought totheforeuniquefeatures inlowReynoldsnumber,compressible flows. For example, [1] numerically and experimentally investi-gatedtheflow pastatriangularairfoilatReynoldsnumbers3000 and 10000 and Mach numbers 0.15 and 0.5. They showed that compressibilitytendstoelongatethewake,causingthetransition from linearto non-linear lift andthe subsequent vortex-induced lifttobedelayedtohigheranglesofattack.[2] demonstrated sim-ilar wakeelongation fortheflowpast atwo-dimensional circular cylinder,belowandabovethecriticalReynoldsnumberofthefirst, Hopfbifurcation. Abovethe criticalReynoldsnumber, thisresults ina largervortex-sheddingwavelength,i.e.alower shedding

Table 1

PropertiesofEarthandMarsatmospheres. Earth Sealevel Mars Ground level Earth Altitude 30 km Gas N2(78%) O2(21%) CO2(96%) N2(78%) O2(21%) Gravity (m.s−2_{) 9.81 3.72 9.71} Density (kg.m−2_{) 1.225 0.014 0.018} Pressure (Pa) 105 _{600 1154} Temperature (K) 288 210 227 Speed of sound (m.s−1_{) 340 238 302}

quency.Wakeelongation was found tobe associatedwithhigher dragand,inthestablestate,withdelayedseparation.

In addition to little knowledge on the flow physics, it is not clear how conventional, numerical tools used to predict aerody-namicperformancearesuitedtolowReynoldsnumber, compress-ible flows.Yet,the designofsmallsizeunmannedvehicles flying inlowpressureenvironmentreliesontheaccuracyofsuch numer-icaltools.

Inthispaper,thisissueisaddressedbycomparingpredictions fromlowandhighfidelity numericaltoolswithexperimental re-sultsobtainedunderextremeconditionswithtypicalReynoldsand Machnumbersintherange

[

100

−

10000

]

and

[

0

.

1

−

0

.

9

]

respec-tively.These numericaltools areapplied to the flow pastairfoils androtorsoperatinginsuch regimes.It isshownthat whilehigh fidelity approaches provide reasonable estimates of aerodynamic forcesforalloperatingconditions,lowfidelityapproachesare lim-itedtolowangleofattack/pitch anglecases,whichstillmakethem suitabletoaerodynamic optimization.Inaddition,uniquefeatures oflow Reynoldsnumbercompressible flowsare revealed, includ-ing,forexample,wakeelongation andsubsequentdampingoflift fluctuations, anddisplacement of shock foot far from the airfoil surface. Finally, experimental data for rotors(which are virtually unavailableintheliterature)arereportedanddeeper insightinto the resulting,enhanced aerodynamic forces(due to leading edge vortexstability)isprovidedbymeansofnumericalapproaches. 2. Numericalapproaches

Inthis section, thetheory and numericalprocedures underly-ing three differentmodels withincreasing complexity are briefly described: (i) two-dimensional potential approaches, (ii) three-dimensionalvortexlatticemethods(VLM)and(iii)numerical sim-ulationsofthetwoandthree-dimensionalunsteadyNavier-Stokes equations(NS).

2.1. Two-dimensionalpotentialflowmethods

Aerodynamic performance of two-dimensional airfoils is first evaluated witha potential flow panel methodcombined withan integral boundary layer formulation. The latter treats both lami-narandturbulentlayersandempiricallydeterminesthetransition point using an eN method. Preliminary linear stability analysis on triangularand camberedairfoils up to a Reynolds numberof 6000wereperformedandshowedthattransitionwastriggeredfor

N

<

1.Therefore,inwhatfollows,theempiricalvalueforN isfixed sufficiently large to avoid transition. Moreover, compressibilityis intrinsicallyaccountedforinthecompressibleboundarylayer for-mulationandappliedasaKarman-Tsiencorrectioninthepotential method.Xfoilcode[3] isusedtoefficientlysolvetheproblemvia

aglobalNewton method.Further detailsonthe numerical proce-durecanbefoundin[3].Thenumberofpanelsusedtodiscretize an airfoilis such that the solution is converged with respect to spatialdiscretizationandisontheorderof160forallcasesshown inthispaper.

2.2. Three-dimensionalvortexlatticemethods

A vortex lattice approach is used to predict the aerodynamic performance ofrotorswithrelativelylowcomputationalcost. Nu-merical simulations are performedusing ONERA’s in-house code PUMA, whichcombinesliftinglineandfree wakemodels[4].The lifting lineapproach relieson two-dimensionalairfoilpolars (ob-tained from resolution of the Navier-Stokes equations, see next section)andapplies3Dcorrectiontoaccountforbladesweepand unsteadyphenomena (e.g.dynamic stall).Thefreewakeapproach reliesonthetheorydeveloped by[5] whichdescribestheunsteady evolution ofa wakemodeled byapotential discontinuitysurface. The blade andwake are discretized using 50 non-uniformly dis-tributedradialstations(squarerootdistributionforincreased res-olutionnearthe tip). Thecomputation isadvanced intime using forwardEulerschemewithatime stepthatcorresponds toa10◦ rotation of the blades. Fourrotations are needed forinitial tran-sientstosufficientlydecay.

2.3. NumericalsimulationsoftheNavier-Stokesequations

The two and three-dimensional unsteady Navier-Stokes equa-tions are numericallysolved usinga finitevolume method. Com-pressibleandincompressibleformulationsareconsidered,typically for Mach numbers above and below 0.2 respectively. Resolution is achievedusing ONERA’sin-housecodeelsA [6] and StarCCM

+

commercialcode [7].Spatial andtemporaldiscretization schemes areofsecondorderinbothcodes,withexplicitandimplicit tem-poralmarchingforelsAandStarCCM

+

respectively.

2.3.1. Airfoilsimulations

For both two-dimensional and three-dimensional airfoil cases atReynoldsnumbersontheorderof103_{andsubsonicMach} num-bers,itwasshownthatatypicalspatialresolutionof



s

/

c

=

0

.

01 allows fortheRichardsonextrapolatedliftanddrag [8] tobe ap-proximatedwithin1

.

5%.Anexampleofgridconvergenceisshown in Fig. 1a where the lift coefficient is displayed as a function of the typical grid spacinginthe wake ofthe airfoil,



x

/

c.2D and 3D resultsareobtainedfora 10◦ angleofattack triangularairfoil [1] at Reynolds and Mach numbers 3000 and 0.15, respectively. Fig. 1b showsQ-criterionisosurfaces colored by spanwise vortic-ity, obtainedon the 3D configurationfor typicalgrid spacings of

c

/

50 (2millioncells)andc

/

160 (70millioncells).Itisshownthat althoughthelargestgridspacingc

/

50 capturesthree-dimensional spanwise instabilities in the airfoilwake (leading to hairpin vor-tices),itisnotabletocapturethemneartheleading edge where the flow exhibits a nearly two-dimensional pattern. As such, it is interesting tonote that forthisspatial resolution,liftobtained from3Dsimulationconvergestowardsthatobtainedfrom2D sim-ulation. Note that the 3D caseshown inFig. 1 corresponds to a 3.3aspectratiowingwithsymmetricalboundaryconditionsatthe tips. Other 3D configurations withnon-slip wallsatthe tipsand witha0.3aspect ratiowingwillalsobe addressedinsection5.1. Subscripts0.3 and3.3 are usedto denoteaspectratio0.3and3.3 respectively.

Fig. 1. Liftcoefficientasafunctionofgridspacingobtainedfor2Dand3Dflowspastatriangularairfoil(a).Q-criterionisosurfacescolored byspanwisevorticitycontours obtainedforgridspacingsx/c=0.02 (top)and0.00625(bottom)onthe3D,aspectratio3.3configuration(b).

Fig. 2. Low pressure experimental facility (a), sketch of the test chamber (b) and of the rotor test bench embedded in the chamber (c). Furthermore, decreasing the time step beyond



tU

/

c

=

0

.

02

doesnotyieldsignificantchangesintheresults.Therefore,inwhat follows,NS results will be shown for



s

/

c

≤

0

.

01 and



tU

/

c

≤

0

.

02. Structured (hexahedral cells) and unstructured (hexahedral trimmed cells) grids are used with elsA and StarCCM

+

respec-tively,resulting ina typical number ofcells on the order of105 for2D cases and2

.

106 and2

.

107 for 3D caseswith 0.3and3.3 aspectratiowing,respectively.

Simulationsarerunfor100convectivetimestoensurethat ini-tialtransientshavesufficientlydecayed.Afterwards,timeaveraged quantitiesareobtainedbyaveraginginstantaneousvaluesover20 convectivetimes.

2.3.2. Rotorsimulations

Forthree-dimensionalrotor cases,similar spatial resolutionto thatusedforairfoilcasesisused(



s

/

c

=

0

.

01)resultingina typ-icalnumberofcellsontheorderof7

.

107.Again,structured (hex-ahedral cells) andunstructured (polyhedralcells) grids are used withelsAandStarCCM

+

respectively.Ontheotherhandthetime stepislarger(



tU

/

c

=

0

.

08),whichismadepossiblebythe quasi-steadynatureoftheflowontherotorconsideredinthispaper(as willbe showninsection 5.3). Alsonote that U isthe velocityat thebladetipsuchthatthebladespanoperatesatvelocitiesbelow

U ,henceatlowernon-dimensionaltimestep.

Simulationsare run for5rotor rotations to ensurethat initial transientshavesufficientlydecayed. Timeaveragedquantities are then obtained by averaging instantaneous values over one rotor rotation.

3. Experiments

3.1. Lowpressurewindtunnel

Experimentaldataby[1] are usedtoassessthevalidity of nu-mericalapproachesforairfoilsoperatinginthelowReynolds num-ber, compressible flow regime. Data are obtainedusingthe Mars Wind Tunnel (MWT) atTohoku University.The MWTconsists of an indraft windtunnelhoused insidea vacuumchamber that al-lowslow-density compressible experiments.It has a 100 by 150 mm test section with typical turbulenceintensity below1%. The airfoilis triangularwith 30mm chordlength and100 mmspan (henceaspect ratio3.33)andamaximumthicknessof1.5mm.A schemeoftheairfoilisprovidedinsection5.1,Fig.3a.Typical op-eratingconditionsleadtoaReynoldsnumberontheorderof103 - 104 _{withMachnumbersontheorderof0}

_.

_{1 - 1.Moredetailson} theexperimentalsetupcanbefoundin[1].

3.2. Lowpressurechamber

Experimental measurements of the aerodynamic performance ofrotorsoperatinginlow pressureenvironmentareconductedin alowpressurechamberof18m3_{volumeatONERAFauga-Mauzac,} Fig.2a.Thetestbenchconsistsofa0.457mdiameterrotor manu-facturedoutofcarbonandrotatedbymeansofaFaulhauber4490 H 024Bbrushless motor,Fig. 2b.The propulsionsetis fixed toa mast on top of whicha 10Newton strain gauge is mounted for thrust measurements. A similar gauge is used at the rear ofthe motorfortorquemeasurements.Thrustandtorquemeasurements

Fig. 3. Triangular airfoil profile (a) and computational domain of the ‘wind tunnel’ configuration (b). are acquiredat a frequencyof 10kHzduring 10seconds,which

ensures statisticalconvergence.In addition, each measurementis performedfourtimesandreportedvaluesareobtainedby averag-ing these four measurements. Data accuracy is derived fromthe maximum deviation of each measurement to the average value ofthe four measurements andfrom discrepancies between mea-surements and calibration weights (under standard, atmospheric pressureconditions).Itwasestimatedtobebelow5% of reported values.

Experimentsareconductedatapressureof2000Pa(minimum achievablepressureis10Pa).BothAirandCO2 (96%) weretested andyielded similar results,in agreement withprevious observa-tions by [9]. Therefore, Air was used forall cases shown in this paper.

4. Datareduction

Resultsare analyzedinterms ofglobalandlocalaerodynamic performanceaswellasflowquantities.

Globalaerodynamicperformanceofairfoilsisassessedusinglift

CL

=

2L

/

ρ

SU2 anddrag CD

=

2D

/

ρ

SU2 coefficients, where

ρ

,U ,

L and D are the fluid density, freestream velocity, lift and drag forcesrespectively.S isthesurfacearea,whichisequaltothewing chordc for2Dcasesandtothewingchordtimeswingspanc

×

b

for3Dcases.Globalaerodynamicperformanceofrotorsisassessed using the thrust CT

=

T

/

ρπ

R2U2 andtorque CQ

=

Q

/

ρπ

R3U2

coefficients,whereR and U aretherotorradiusandbladetip ve-locityrespectively.

Localaerodynamicperformanceofairfoilsisassessedusingthe pressurecoefficientCp

=

2

(

p

−

p∞

)/

ρ

U2,wherep andp_∞arethe dimensionalstaticpressureonandfarupstreamtheairfoil, respec-tively.Local aerodynamic performance ofrotorsis assessedusing the sectional thrust coefficient CsT

=

sT

/

ρ

SU2, where sT is the

sectionalpressure thrustand S is thearea ofthebladespanwise section(orbladeelement).

Finally,flowfeaturesare displayedusingnon-dimensional vor-ticity

ω

∗

=

ω

c

/

U , Q-criterion and pressure. Their corresponding oscillating frequencies f are non-dimensionalizedusingthe wing chordandfreestreamvelocitiesSt

=

f c

/

U .

5. Results

5.1. Evaluationofnumericalmethodsontriangularairfoil

Theliftanddragcoefficientsandthelift-to-dragratioobtained onatriangularairfoilusingbothpotentialflow(Xfoil)and Navier-Stokes(elsAandStarCCM

+

)solversarefirstcomparedwith exper-imentalandnumericalresultsfrom[1].Thepotentialflowsolution istwo-dimensionalwhilesolutionstotheNavier-Stokesequations are presented for2D and 3D cases. 3D cases with aspect ratios 0.3and 3.3are considered. These aspect ratios are chosen to be consistent with numerical simulations (aspectratio 0.3) and ex-periments (aspect ratio 3.3) from [1]. An additional case taking intoaccountwindtunnelwallsisalsosimulated.Fig.3illustrates the airfoil profile and the aspect ratio 3.3, 3D case with wind

tunnel test section. Note that this particular airfoil is here con-sidered precisely because both numerical and experimental data incompressible,low Reynoldsnumberflowconditionsarereadily available intheliteratureforcomparison[1].[1] selectedthis tri-angularairfoilasapotentialcandidateforthedesignofpropellers forMartianaircraftsduetoitssimplegeometry,withsharpedges andflatsurfaces.Thesecharacteristics(i.e.sharpleadingedgesand flatsurfaces) werepreviouslyfound topromoteaerodynamic per-formance in the low Reynoldsnumber compressibleflow regime [10].

Fig. 4a compares two-dimensional numerical results with ex-perimental results forReynolds number 3000and Mach number 0.5. Reasonable agreement between numerical and experimental data are observed at low angles of attack, in the linear regime where theflow is mostly attached.As

α

increasesandflow sep-arationbecomessignificant,beyond

α

≈

5◦,theliftcurveobtained from 2D Navier-Stokes computations progressively diverge from theexperimentalcurve andeventuallyreacheslargediscrepancies as theflow fullyseparates fromthe airfoil, beyond

α

=

11◦. The drag curveexhibitsa rathersimilartrend,yetwithbetter approx-imation ofexperimentaldata intherange

α

∈ [

5◦

−

11◦

]

.Onthe other hand, liftanddrag obtainedfromthe potential flow solver remains consistent with experimental data up to large angles of attack.

Figs.4band4c comparetheliftobtainedfrom3DNavier-Stokes computations with that obtained from experiments. Fig. 4b fo-cuses onsimulations withspanwiseextent 0.3chord(referred to as3D0.3 NS).Recallthat symmetricalboundaryconditionsareset at the wingtips which makes the configuration nominally two-dimensional butallows three-dimensional (spanwise) instabilities to developandaltertheoverallflow structure.Again,itisshown that both liftanddrag obtainedfrom3D Navier-Stokes computa-tions areinreasonable agreementwithexperimentaldataforlow anglesofattack.Inaddition,thereisnoobservabledifferences be-tween 2D and 3D Navier-Stokes computations for

α

<

11◦. This suggeststhat three-dimensional effectsare weak inthis rangeof angleofattackandisconsistentwiththefactthattheflowisnot yetfullyseparated.Beyond

α

=

11◦,massiveseparationoccursand promotesthree-dimensionalspanwiseinstabilities.Hence,bothlift and drag coefficientsare reduced withrespect tothose obtained from 2D Navier-Stokes computations. Yet, despite this reduction, 3D Navier-Stokescomputationswithspanwiseextentequalto0.3 chordcannotrecoverexperimentalresults.

Fig.4cfocusesonsimulationswithspanwiseextent3.3chords (referredtoas3D3.3 NS).Here,bothsymmetricalandwall bound-ary conditions at the wing tips are considered. Comparing with previousresults,itisshownthatforsymmetricalboundary condi-tions,thespanwiseextentofthecomputationaldomainhasno sig-nificantinfluence onaerodynamic performance,atleastfor span-wise extents above 0.3chord. In other words, a spanwise extent of0.3chordissufficienttocapturethree-dimensionalinstabilities andtheirinfluenceonaerodynamicloadsathighanglesofattack. Thisisconsistentwithresultsfrom[1].Conversely,takinginto ac-count wall boundary conditions hasa significant impact on both liftanddragpredictionsandresultsinamuchmoreaccurate esti-mationforallthreeanglesofattacktested.

Fig. 4. Comparisonofliftanddragcoefficientsandlift-to-dragratiosobtainedfromnumericalapproachesandexperimentsonatriangularairfoilatReynoldsandMach numbers3000and0.5respectively.2Dnumericalapproaches(a)and3Dnumericalapproacheswithspanwiseextent0.3(b)and3.3(c).

Overall,these results suggest that 3D effects arising fromthe windtunnel test sectiontend tolimit flow separation.Therefore, 2DNavier-Stokescomputationsand3DNavier-Stokescomputations with symmetrical boundary conditions predict early separation, whichleads tolift coefficientsrapidlydivergingfrom experimen-talvalues.Onthecontrary, because2D potentialflow solvers fail to predict massive flow separation (at least without special em-pirical treatment), the potential flow solution converges towards a partially attached flow even at high angles of attack. That is, the solution is somehowsimilar to that arising from 3D effects inducedbywallboundaryconditions.Therefore,despitethe seem-inglyaccuratepredictionofaerodynamic loadsoverthefullrange ofanglesofattacktested,potentialflowtheoryshouldnotherebe

viewedasan accurateapproachforhighanglesofattack aerody-namics.

Furthermore,because3DNavier-Stokescomputationswithwall boundaryconditionspredictexperimentaldatawithreasonable ac-curacy,itcanbeinferredthat3DNavier-Stokescomputationswith symmetrical boundary conditions are suited to the prediction of aerodynamic loadson anominallytwo-dimensional configuration atlargeangles ofattack. Buildingonthat, thesimilaritybetween 2D and3Dnumerical resultsatlowangles ofattack suggestthat all 2Dmethodsare suitedtothepredictionofaerodynamicloads when the flow is mostly attached, i.e. when three-dimensional effectsare limitedandwheretheefficiencyoftheairfoilis max-imum. Adirect outcome isthat potential flow theory, aswell as 2DNavier-Stokescomputations,canbeusedwithreasonable

accu-Fig. 5. Liftanddragcoefficientsandlift-to-dragratiosobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatReynoldsnumber3000andMachnumbers0.1, 0.5,0.7and0.8.

racyforairfoilshapeoptimizationinthecontextoflowReynolds numbercompressibleflows.

5.2. ReynoldsandMachnumbereffectsoncamberedairfoil

In light of the above conclusions, two-dimensional Navier-Stokes simulations of the flow past a cambered airfoil are per-formedtohighlightReynoldsandMachnumbereffectson aerody-namicperformanceinthelowReynoldsnumbercompressibleflow regime.Theairfoilisacircularcamberedplateandhas6

.

35% cam-ber and1% thickness with sharp leading edge andblunt trailing edge andwas previously found to exhibit relatively good perfor-manceatlowReynoldsnumber[11].Computationsareperformed forarangeofanglesofattack

α

∈ [

0◦

−

15◦

]

(withastepof1◦),for Reynoldsnumbers100,1000,3000and10000andMachnumbers 0.1,0.5,0.7,0.8and0.9.

5.2.1. Machnumbereffects

Fig.5 showsthe liftand drag coefficientsandthe lift-to-drag ratio obtained at Re

=

3000 for Mach numbers 0.1, 0.5, 0.7 and 0.8. First,itis observedthat liftincreaseswithMach numberfor a broad range ofangles of attack tested, i.e.

α

∈ [

2◦

−

10◦

]

. Yet, thetrendis oppositebelow2◦ andabove 13◦.Thispoint willbe discussed in the next paragraph. Second, it is shown that drag increases withMach number ina more significant amount than lift, leading to a decrease in lift-to-drag ratio. Finally, it can be observedthattheangleofattack correspondingtomaximum lift-to-dragratiodecreaseswithMachnumber.

To provide insight into the mechanisms responsible for these trends, Fig. 6 displays spanwise vorticity contours obtained for Mach numbers 0.1, 0.5 and 0.8 at Reynolds number 3000. For

α

=

2◦,theflowseparatesnearthetrailingedgeandtheupperand lower surface, opposite sign vorticity layers interact at the trail-ingedgeandrollupintooppositesignvorticalstructures,leading to avon Kármán vortexstreet. Alternatively, one can understand theemissionoftrailingedge vorticesastheunsteadyresponseof boundcirculationtotheemissionofvorticesinducedbyseparation on the upper surface. It can be seen that the width of the vor-texstreetslightlyincreaseswiththeMachnumber,indicatingthat theseparationpointmovesupstream. ThecorrespondingStrouhal number decreases from 2.17 to 2.07 and 1.79 at Mach 0.1, 0.5 and0.8respectively(seeappendixforfurtheranalysisonunsteady responseof forcesandmoments tovortex shedding).Similar ob-servationscanbemadeat

α

=

5◦and8◦.Atthispoint,theeffect ofincreasingtheMachnumberatagiven

α

canthusbeviewedas similartoincreasing

α

atagivenMachnumber.Thistends to in-creaselift(intherangeof

α

considered),whichsupportsprevious observationsonlifttrend.

For Mach

=

0.1and

α

=

10◦ the uppervorticity layer interacts withtheuppersurface oftheairfoilandrollsupintoaclockwise rotating vortex before reaching the trailing edge. This ‘early’ in-teraction leadstoanupward deflectedwake.Thisphenomenonis more clearly visibleat

α

=

12◦.ForMach

=

0.5, such a wake pat-tern isnot observedat

α

=

10◦ butappears athigher

α

,seefor example

α

=

12◦ and15◦.Conversely, theflow atMach

=

0.8still exhibitsanon-deflectedwakeupto

α

=

15◦.Forthelatter,itcan beseenthattheuppervorticitylayerdoesnotrollupintoavortex priortointeractingwiththetrailingedge,althoughtheseparation point appears very close to the leading edge. A direct outcome is thatincreasing theMachnumberreducesthe fluctuatingloads on theairfoil. Forinstance, therelative standarddeviation ofthe

α

=

15◦liftcoefficientisequalto0.104,0.098,0.039and0.019for Machnumbers0.1,0.5,0.7and0.8respectively(seeappendix).The absenceof‘early’vorticitylayerrollupcanbecorrelatedwiththe absenceofrapidincreaseintheliftversus

α

curve,whichexplains why liftincreasesasthe Machnumber decreases forthehighest valuesof

α

tested.

The time-averaged distribution of pressure coefficients Cp on

Fig. 7 further highlights the vorticity roll up mechanism and its influence on aerodynamic loads at

α

=

12◦. It isshown that the

Cp distributionontheuppersurfaceoftheairfoilisratherflatat

M

=

0

.

8,wherenorollupisobserved.Ontheother hand, vortic-ity roll up at M

=

0

.

1 induces a bump in Cp distribution, near

x

/

c

=

0

.

6. This bump is followed by a drop at the rear of the airfoil wherevortices are shedand advected into thewake. This specific bump-drop pattern resemblesthat induced by a laminar separation bubble butis here the time-averagedfootprint of the unsteady formation and shedding of clockwise rotating vortices. Fig. 7alsoshowsthe Cp distribution atM

=

0

.

9. Itisinteresting

tonotethatthedistributionissimilartothatobtainedatM

=

0

.

8 although the flow is transonic (see Fig. 8). That is, it is shown that the presence of shocks do not significantly affect the pres-sure distributionin thislow Reynolds numbercompressible flow regime.Here,enhanced viscouseffects(i.e.smooth velocity gradi-entsassociatedwiththickboundary/shearlayers)tendtomovethe shockfootawayfromtheairfoil, whichisauniquefeatureoflow Reynolds numbercompressibleflow regimes.In other words, ex-tensively investigatedphenomenon suchasshockwave boundary layerinteractionmaynotapplyhere,resultingincompletely differ-entresponsesofaerodynamicperformancetotransonic/supersonic regimes.

5.2.2. Reynoldsnumbereffects

Fig. 9 showsthelift anddrag coefficients andthe lift-to-drag ratioobtainedatMach

=

0.5forReynoldsnumbers100,1000,3000 and 10000. Thetrends are in linewith existing literature (under

Fig. 6. Instantaneousspanwisevorticityflowfieldsobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatReynoldsnumber3000andMachnumbers0.1,0.5 and0.8.Snapshotsareshownforsixanglesofattackα=2◦,5◦,8◦,10◦,12◦and15◦.

Fig. 7. Distribution oftime-averaged pressure coefficients Cp obtained from 2D

Navier-StokescomputationsforMachnumbers0.1,0.8and0.9.

incompressibleconditions,e.g.[12])demonstratingadecreaseand anincreaseinliftanddragcoefficientswithReynoldsnumber, re-spectively.Theincreaseindragcoefficientisparticularlysevereas theReynolds numberis decreased from1000to 100, where vis-cous drag becomes dominant. As a consequence, the lift-to-drag coefficientsignificantlyincreaseswithReynoldsnumber,beingan

Fig. 8. Instantaneousspanwisevorticityflowfieldsobtainedfrom2DNavier-Stokes computationsforacamberedairfoilatReynoldsnumber3000andMachnumber 0.9. Snapshotsareshown forthree anglesofattackα=2◦,8◦ and 12◦. Trans-parencyisappliedtovorticitycontours,onwhichcontours ofpressure gradient magnitudearesuperimposedtohighlightshockwaves.

orderofmagnitudegreateratRe

=

10000thanatRe

=

100.In addi-tion,itcanbeseenthatthemaximumlift-to-dragratioisobtained atlargeranglesofattackastheReynoldsnumberisdecreased.The dependence ofaerodynamicperformanceonReynoldsnumbercan hereagainbe correlatedwiththevorticity flowfieldsdepictedin Fig.10.

Fig. 10 displays spanwise vorticity contours obtained for Reynoldsnumbers100,1000and10000 atMach number0.5. For

Fig. 9. Liftanddragcoefficientsandlift-to-dragratiosobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatMachnumber0.5andReynoldsnumbers100, 1000,3000and10000.

Fig. 10. Instantaneousspanwisevorticityflowfieldsobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatMachnumber0.5andReynoldsnumbers100, 1000and10000.Snapshotsareshownforsixanglesofattackα=2◦,5◦,8◦,10◦,12◦and15◦.

α

=

2◦,theflowatRe

=

100ischaracterizedbytwothick,opposite signvorticitylayersontheupperandlowersurfacesoftheairfoil, exhibitingasteadypattern.TheflowatRe

=

1000israthersimilar, yetwith thinnerandstronger shear layers. AsRe is increasedto 10000,theupperandlowershearlayers interactnearthetrailing edge,leadingto anunsteadywakecharacterized byopposite sign vorticalstructures. Theseflowpatternsarequalitativelysimilar at

α

=

5◦.

As

α

is increased, the flow at Re

=

100 remains roughly un-changed. Conversely, it becomes unsteady at Re

=

1000, with the shearlayersinteractingnearthetrailingedge.

At Re

=

10000 and

α

=

5◦, theupper shearlayer rolls up into aclockwiserotatingvortexnearmid-chord.Thelatterisadvected towards the trailing edge where it eventually interacts with the lowershearlayer.At

α

=

10◦,therollupoftheuppershearlayer is more severe,which leads to a strong interaction withthe up-persurfaceandthesubsequentdeflectedwakepattern(previously described). While the interaction betweenthe clockwise rotating vortex andtheuppersurface oftheairfoilatRe

=

10000becomes stronger as

α

is further increased, i.e. as the separation point movesupstream,theuppershearlayeratRe

=

1000isimmune to rollupbeforeinteractingwiththelowershearlayeratthetrailing

Fig. 11. Comparison of thrust and torque coefficients and thrust-to-torque coefficients ratio obtained from numerical approaches and experiments on a two-bladed rotor.

Fig. 12. Surface pressure contours and Q-criterion isosurfaces obtained from 3D Navier-Stokes computations for a two-bladed rotor with pitch angle 15◦, 19◦, 25◦and 30◦.

edge.Therefore, whilethe wakeat Re

=

1000exhibitsa relatively simplepatterncharacterized by alternate vortices up to

α

=

15◦, theflowatRe

=

10000transitstoachaoticstate(seealso fluctuat-ingforcesandmomentsintheappendix).

TherollupoftheuppershearlayeratRe

=

10000explainsthe suddenincreaseinliftobservedat

α

=

8◦onFig.9.Conversely,the robust shear layers at Re

=

100and 1000explain the smooth in-creaseoftheliftcoefficientwith

α

.Thatis,atthelowestReynolds numbers,thereisnodrasticchangesintheflowas

α

isincreased, leadingto arelatively simplerelationbetweenliftandtheairfoil projectedareac

×

sin

α

.

Overall, the present results show that, in the low Reynolds numbercompressibleflowregime,bothReynoldsandMach num-bershavesignificantimpactontheflowpatternandtheresulting aerodynamic performance. In particular, changes in aerodynamic performanceduetovariationsinReynoldsandMachnumbersare onthesameorderofmagnitude.Therefore,foragiven character-istic dimension (e.g. wing chord) and environing pressure, there existsanoptimaloperatingspeed(i.e.anoptimalRe-Machcouple) thatleadstooptimalaerodynamicperformance.Thisis fundamen-tallydifferenttoconventionalaerodynamicswherelargeReynolds andMachnumbereffectsareusuallyuncoupled.

Fig. 13. Sectional thrust coefficient as a function of the non-dimensional rotor radius obtained from 3D NS computations and VLM for pitch angles 15◦, 19◦, 25◦and 30◦.

5.3. Evaluationofnumericalmethodsontwo-bladedrotor

Inthissection,thecomparisonbetweennumericalapproaches andexperimentaldatais extendedtothe flowpasta two-bladed rotoroperating underhoveringconditions.Basedon theprevious resultsthat2D numericalapproachesaresuitedtotheprediction of aerodynamic loads in the low Reynolds number compressible flowregime, aroundmaximumefficiency,3D vortexlattice meth-odsarehereappliedusingpolarsfrom2DNavier-Stokes computa-tionsandcomparedto3DNavier-Stokes computationsand exper-iments (Fig. 11). The blades consist of a 6

.

35% camber, 1% thick airfoil (see section 5.2) with constant chord and constant twist angle

β

along the span. It is similar to that tested in [13]. The rotation speed and ambientpressure are such that the Reynolds andMach numbersatthebladetipare ontheorderof6000and 0.35respectively.

Fig.11showsarelativelygoodagreementbetweenNSandVLM approaches.Thesenumericalmethods,however,overestimate both experimentalthrustandtorquecoefficients.Discrepanciesinthrust and torque predictions somehow compensate each other, lead-ing to a fairapproximation ofthe thrust-to-torque ratio. Despite these quantitative discrepancies, the trends in thrust andtorque versus pitchangle arequalitatively similar fornumerical and ex-perimentalapproaches. Overall,CT increasestoamaximumvalue

at

α

≈

30◦ and CQ continuously increases with

α

. As a

conse-quence, CT

/

CQ exhibits an optimal value, which is found to be

around10◦pitchangle.

AcloserlookattheNSandVLMcurvesrevealsclosermatch be-tweenboth approachesatlow pitchthanathighpitchangles.At

highpitchangles,resultsobtainedfromNavier-Stokessimulations slightlyoverestimateVLMresults.This‘extra-lift’obtainedwithNS simulations may arise from the development of a stableleading edge vortexonthe uppersurfaceofthe rotorblades. This intrin-sicallythree-dimensionalmechanismcannotbepredictedbyVLM. Fig.12 displaysiso-surfaces ofQ-criterionobtainedfromNS sim-ulations for pitch angles 15◦, 19◦, 25◦ and 30◦. At higher pitch angles (

β >

20◦), itisshownthat theflow separatesatthe lead-ing edge and rolls up into a conical leading edge vortex (LEV). On the contrary to the LEV that developson a two-dimensional airfoiland thateventually sheds into thewake, theLEV here re-mainsstablyattachedtotheblade,inducingasustainedlow pres-suresuction regionon itsupper surface,which inturngenerates thrust.StabilityoftheLEVisacommonfeatureinthe aerodynam-icsoflowaspectratioflappingandrevolvingwings(whichoperate ina comparablerangeofReynoldsnumbers)andishypothesized to resultfromrotationalaccelerations,makingit afundamentally three-dimensional phenomenon [14,15]. Because of this stability, sectionalaerodynamic forcesontheinboardpartofthebladeare steady (in contrast to those on two-dimensional airfoils, see ap-pendix).Intheoutboardregionoftheblade,i.e.nearthetip,itcan beobservedthat theLEVburstsintosmallerscalestructuresasit merges withthetipvortex.Thisphenomenon isalsoinlinewith previousnumericalandexperimentalobservationsatlowReynolds numbers(e.g.[16–18]).

Fig.13comparesthesectionalthrustdistributionalongthe ro-torbladeobtainedfromNScomputationswiththatobtainedfrom VLM.Whilereasonableagreementbetweenbothapproachesis ob-served upto25◦ pitchangle,non-negligiblediscrepanciesare

ob-Fig. 14. Fluctuatingpartsoflift,dragandpitchingmomentcoefficientsobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatReynoldsnumber3000and Machnumbers0.1,0.5and0.8.

served at 30◦ pitch angle where the effectof the LEV on thrust ismaximum.Discrepanciesobservedintherange0

.

2

<

r

/

R

<

0

.

8 arepartlycompensatedforby discrepanciesatthetip wherethe dynamicsofthevorticalstructures(merging betweenLEVandtip vortex)isalsoverycomplexandcannotbecapturedthroughVLM. Assuch,despitetherelativelygoodagreementobservedonFig.11, VLMshould not beviewed asan accuratemethod forhighpitch angleswhereseparationoccursattheleadingedge.Notehowever thatspecialtreatmentofthetwo-dimensionalpolarsusedinVLM canbeaddedtoaccountforrotationaleffectsonLEVstability[19]. Yet,becausemaximumefficiency isobtainedatlow pitchangles, presentresultssuggestthatVLMissuitedtotheaerodynamic op-timizationofrotorsinthelowReynoldsnumbercompressibleflow regime.

6. Conclusion

Flight on Mars and in the stratosphere, high speed trains in low pressure tubes andliquidatomization share unique atmody-namicfeaturesinthatthey involvebodiesmovingina compress-ible viscousflow, i.e.theyoperateina low-to-moderateReynolds number, compressibleflow regime. Because of thevery prospec-tivenatureoftheseapplications,verylittleknowledgeisavailable on thephysics of lowReynolds number,compressible flows. Fur-thermore,itisnotclearwhetherconventionalnumericalmethods usedtopredictaerodynamicforcesonmovingbodiesaresuitedto suchflowregime.

In this paper, the accuracy of low and high fidelity numer-ical approaches in the prediction of aerodynamic forces at low

Fig. 15. Fluctuatingpartsoflift,dragandpitchingmomentcoefficientsobtainedfrom2DNavier-StokescomputationsforacamberedairfoilatMachnumber0.5andReynolds numbers100,1000and10000.

ReynoldsnumbersandmoderatesubsonicMachnumberswas as-sessed.Assessmentwas achievedby comparingnumericalresults withexperimental dataonairfoils from[1] and withpresent ex-periments on rotors. It was shownthat highfidelity approaches, i.e. numericalresolution of the 3D Navier-Stokes (NS) equations, provideresultsingenerallygoodagreementwithexperimentsfor thewholerangeofoperatingconditionstested.Ontheotherhand, it was shown that numerical resolution of the 2D Navier-Stokes equations,potential flow approachesand VortexLattice Methods (VLM)providereasonable estimationofaerodynamicforceswhen theangleofattack/pitch angleislowandtheflowisattached(or weaklyseparated).Forhighanglesofattack/pitch angles,theselow fidelityapproachesfailtopredictmassiveflowseparationand sub-sequent3D effects,sometimesdespitereasonable agreementwith

experimentaldata(whichwasshowntoresultfromcompensating errors). Theseresultsindicated that,in thelow Reynoldsnumber compressibleregime,airfoilandrotoroptimizationcanbeachieved withreducedordermodels.

Inaddition,theinfluenceofReynoldsandMachnumbereffects on the flow past airfoils was analyzed and significant impact on flow separationandsubsequentwakepatternswasdemonstrated. In particular,results indicated thatcompressibilitytends to elon-gatetheairfoilwakeandprevent earlyroll-upoftheuppershear layer, inlinewithpreviousobservations by[1,2]. Thisresultedin a re-orientation of the incompressible deflected wake and to a smootherlift-to-angle-of-attackcurveandadampingoflift fluctu-ations.Moreover,uniquefeaturesassociatedwiththedisplacement of theshockfootaway fromtheairfoilsurface were revealedfor

transonic operatingconditions.Ontheotherhand,Reynolds num-bereffectswerefoundtobequalitativelysimilartothosereported inextensiveworksatlow Machnumbers.Overall, itwas demon-stratedthatinthelowReynoldsnumbercompressibleflowregime, theimpact ofboth ReynoldsandMach numberson aerodynamic performanceare onthesameorder ofmagnitudewhich suggests thatforagivencharacteristicdimensionandenvironing pressure, thereexistsanoptimaloperatingspeedthatleadstooptimal aero-dynamicperformance.

Finally, experimental data for rotors operating in the low Reynoldsnumber compressible regime were provided and, along withnumerical results, similar flow patternto that observed on bio-inspiredflappingandrevolvingwings(i.e.stabilityofthe lead-ingedgevortexandenhancedthrustproduction)wereanalyzed.

TheseresultsprovidedinsightintothephysicsoflowReynolds number compressible flows and into the accuracy of numerical approachesfor their prediction, thereby helping futuredesign of Marsandstratosphericflyingvehicles.

DeclarationofCompetingInterest

Wedeclarethatwehavenocompetinginterest. Acknowledgements

The authors greatefully acknowledge the Centre National d’E-tudes Spatiales (CNES) for partly funding this work. They are alsothankful toRémy Chanton,Sylvain Belliot,Henri Dedieuand ValérieAllainfromISAE-SUPAEROfortheirtechnicalsupport. Appendix A. Unsteadyforcesandmomentsoncamberedairfoil

Unsteadyresponses oflift,dragandpitchingmomentobtained ona camberedairfoilat Reynoldsnumber3000 andMach num-bers0.1, 0.5and0.8arefurther analyzedforangles ofattack 2◦, 8◦ and 15◦. Fig. 14 plots the instantaneous fluctuating parts of lift,dragandpitchingmomentcoefficients(C_L,C_D andC_m respec-tively)asafunctionoftheconvectivetime.Notethatthepitching momentcoefficientisdefinedasCm

=

2m

/

ρ

ScU2,wherem isthe

pitching moment about a spanwise axis located a quarterchord downstreamoftheleadingedge.

At

α

=

2◦,itisseenthatamplitudesofforceandmoment fluc-tuationsdecreaseastheMachnumberisincreasedfrom0.1to0.5 butthenincreaseslightlyastheMachnumberisfurtherincreased to0.8. Asobserved insection 5.2,thisnon-monotonic trendmay resultfromtwocompetingmechanisms.Ontheonehand, increas-ingtheMachnumberatagivenReynoldsnumbertends tomove theseparation point upstream,hence a larger portionof the air-foilisundertheinfluenceoftheunsteady,separatedleadingedge shearlayer.Ontheother hand,it tendstostabilizetheseparated leading edge shear layer which seems immune to ‘early’roll-up. Athigheranglesofattack

α

=

8◦ and15◦,thefirstmechanismis relatively weaker since separation already occurs close orat the leadingedgeevenatlowMachnumbers.Henceforceandmoment fluctuationsmonotonicallydecreasewithMachnumberasaresult ofthestabilizationoftheleadingedgeshearlayer.

Inaddition,itisshownthattheStrouhalnumbergenerally de-creaseswithincreasing Mach number.At

α

=

2◦, St

=

2

.

17, 2.07 and1.79forMachnumber0.1,0.5and0.8respectively.At

α

=

8◦,

St

=

1

.

78,1.65and1.33forMachnumber0.1,0.5and0.8 respec-tively.At

α

=

15◦, theStrouhal numberfirst decreasesfrom0.75

to0.59astheMachnumberisincreasedfrom0.1to0.5,andthen increasesto0.78asthe Machnumberisfurtherincreasedto0.8. Theprecisereasonforthisnon-monotonictrendisunclearatthis point but it should be noticed that for such high angles of at-tack,strongnon-linearitiesmaytriggeradditionalinstabilitiesthat mayaffectthewholesheddingprocess.Inparticular,althoughnot shownhereforthesakeofconciseness,the

α

=

15◦ caseatMach number 0.8 exhibits a low-frequency modulation of the aerody-namic loads, about 27 times lower that the fundamental vortex sheddingfrequency.

Instantaneous fluctuating parts of lift, drag and pitching mo-mentcoefficientsatMachnumber0.5andReynoldsnumbers100, 1000and10000aredisplayedonFig.15.Itcanbeseenthat,forall threeanglesofattack

α

=

2◦,8◦and15◦,increasingtheReynolds numbertendstoincreasebothamplitudesandfrequenciesof fluc-tuations.Inparticular,andaccordinglytoresultsshowninFig.10, forceandmomentsignalsaresteadyatRe

=

100 andmayexhibit chaoticfluctuationsastheReynoldsnumberisincreasedto10000. References

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