Optimization of Wind Turbine Airfoils Subject to Particle Erosion

23 Pages·2015·0.72 MB·English

AIAA Aviation 2015-3393
22-26 June 2015, Dallas, TX
33rd AIAA Applied Aerodynamics Conference

Optimization of Wind Turbine Airfoils Subject to Particle
Erosion
GiovanniFiore∗ andMichaelS.Selig†
UniversityofIllinoisatUrbana-Champaign,DepartmentofAerospaceEngineering,Urbana,IL61801
An optimization of wind turbine airfoils subject to particle erosion is investigated. The erosive
damage to the surface was represented by sand grains colliding with the blade leading edge. The
optimizationwasperformedbyusingageneticalgorithmapproach(GA).Anewly-writtenGAcode
wascoupledwithatwo-dimensionalinviscidflowfieldsolver(XFOIL)andwithaparticledynamics
code (BugFoil). The chosen scheme of the GA was based on random tournament selection. Each
geometrywasevaluatedthroughafigureofmeritthatrepresentedtheairfoilfitnesstodamageand
toaerodynamicperformancewhentheairfoilisdamaged. Multipleapproachestotheoptimization
weretaken. Theinitialstepofthisstudyinvolvedoptimizationthroughgeometryperturbationsof
theleadingedgebymeansofBeziercurves. Itwasfoundthattheleadingedgeshapemodifications
of existing airfoils increase the fitness of the airfoils subject to erosion. The second approach in-
volvedacrossoverofexistingairfoilgeometriesalongwithleadingedgeshapemutations,inorder
to expand the design space. The fitness of such airfoils was improved with respect to the first ap-
proach. However,forboththeseapproachestheairfoilaerodynamicperformancewasdisregarded,
as they were intended to give insight on the optimal airfoil shapes. The final step of the study
made use of an inverse airfoil design method (PROFOIL) to widen the design space even further.
In this approach the aerodynamic performance of the airfoil was evaluated, and the GA became
a two-objective optimization process (maximum lift-to-drag ratio, and erosive figure of merit). It
was observed that bulbous upper leading edges, and slanted, flat lower leading edges allowed for
best erosion performance. Moreover, it was seen that the airfoils with a positive camber required
asmallerangleofattacktooperateatthesameliftcoefficient,hencereducingtheerosivedamage
experienced for sand grain impacts. The airfoils obtained from the inverse design GA algorithm
werecharacterizedbyalift-to-dragratiogreaterthan310foraReynoldsnumberof5.84×106.
Nomenclature
AK = particlenondimensionalmass
c = airfoilchordlength
COE = CostofEnergy
C = airfoildragcoefficient
d
C = particledragcoefficient
D
C = airfoilliftcoefficient
l
d = particlediameter
D = particledragforce
E = erosionrate
FOM = figureofmerit
GA = geneticalgorithm
GAEP = GrossAnnualEnergyProduction
K = erosionrateconstant
∗GraduateStudent(Ph.D.),DepartmentofAerospaceEngineering,104S.WrightSt.,AIAAStudentMember.
†AssociateProfessor,DepartmentofAerospaceEngineering,104S.WrightSt.,AIAAAssociateFellow.
http://www.ae.illinois.edu/m-selig
1of23
AmericanInstituteofAeronauticsandAstronautics
Copyright © 2015 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
LE = leadingedge
m = particlemass
n = erosionratevelocityexponent
r/R = bladesectionradiallocation
Re = freestreamReynoldsnumber
t = time
t/c = airfoilthickness-to-chordratio
U = chordwiseflowfieldvelocitycomponent
V = chord-normalflowfieldvelocitycomponent
V = freestreamvelocity
∞
V = particleimpactvelocity
imp
V = particleslipvelocity
s
x = chordwisecoordinate
y = chordwise-normalcoordinate
α = angleofattack
α = relativeanglebetweenflowfieldandparticlevelocity
r
θ = impactangle
τ = nondimensionaltime
ω = weightusedtocomputeFOM
Subscriptsandsuperscripts
0 = initialstate
l = lowerlimit
P = particle
tr = transition
u = upperlimit
I. Introduction
Windturbinesusedforeletricalpowergenerationaresubjecttofoulinganddamagebyairborneparticlestypicalof
theenvironmentwherethewindturbineoperates. Throughoutthe20-yearlifespanofawindturbine,particlessuchas
rain,sand,hail,insects,andicecrystalsaremajorcontributorstoadeteriorationinturbineperformancethroughlocal
airfoilsurfacealterations.1–6 Windturbinebladesaccumulatedirtespeciallyinthesurroundingsoftheleadingedge.
Inaddition,temperaturejumpsandfreeze-thawcyclesmaycausesmallercoatingcrackstopropagate,promotingcoat-
ing removal and eventually delamination and corrosion damage due to exposure of the internal composite structure.
Theoriginallysmoothsurfaceofthebladesmaychangeconsiderably,andtheincreasedroughnesswillcauseadrop
ingrossannualenergyproduction(GAEP)andanincreaseincostofenergy(COE).7–13
Moderntrendsinthewindturbinemarkethaveshownthebenefitsofoffshoremegawatt-scalewindturbineinstal-
lations14,15inordertomaximizeGAEPwhilereducingCOE.However,offshorelocationsaresubjecttomoreintense
sand erosion than the majority of land installations.13,16,17 Airborne sand particles collide with the blade and cause
micro-cuttingandplowinginthecoatingmaterial18,19 resultinginsurfaceabrasion.20,21 Suchdamageisparticularly
prominentattheoutboardsectionsofthebladewherethelocalrelativevelocityislargercomparedwithinboardsec-
tions.16,22
The weather conditions of a given wind farm site may vary substantially throughout the seasons. Throughout
the world, another common damage scenario is represented by water-based particles, namely raindrops and hail-
stones.23–25 Whenintenseprecipitationoccurs, largeraindropsmayimpactwiththewindturbinebladesandpoten-
tiallypromoteinternalpaneldelamination.23,26 Evensmallerraindropsincreasethemechanicalfatigueinthecoating
plasticmaterials27,28whichtranslatesintosurfacemicro-crackingandafinalerosiveeffect.29
Thewindturbineindustryhasdevelopedseveraldevicestohelpmitigatetheeffectofairborneparticlesstriking
onthebladesurface. Inparticular,chemicalcompanieshavecommercializedleadingedgetapeproductswhichallow
foraconvenientsolutionthatisintegratedintotheblademaintenancecycle.11,30,31 However,fromthestandpointof
2of23
AmericanInstituteofAeronauticsandAstronautics
anaerodynamicdesignersuchanapproachmayappearasalaterfix. Hence,itisvaluabletoinvestigatewhichblade
sectiongeometryfeaturesallowforaminimizationofthesurfaceerosivedamage.
The recent research has shown that the blade damage scenario is complex and multi faceted.22–25,32 From an
aerodynamicstandpointitmustbenotedthatthereexistsacorrelationbetweenthebladeflowfieldandtheparticlein-
teractionwiththebladesurface. Inparticular,smallandlightweightparticlesaremoresusceptibletotheaerodynamic
flowfieldthanbiggerandheavierparticles. Therefore,abladeshapeoptimizationtominimizethedamageofanytype
ofparticlemaybedifficulttoimplement. Forexample,insectsrepresentacomplexscenariowhichischallengingto
undertakeduetothelargevarietyofspeciesandknowledgegapsconcerningtheiraerodynamiccharacteristics. Other
heavyparticlessuchashailstonesarecharacterizedbyatrajectorythatisfairlyindependentofthebladeflowfield.23,25
Littlemaybedonefromashapeoptimizationstandpointtominimizethedamagetothepanelduetohailstonestrikes.
However,thetwoparticlesthataremostsensitivetothebladeflowfieldaresandgrainsandraindrops.22,25Bothparti-
clesareassociatedwithanerosionrate,henceallowingforanestimateofsurfacematerialloss. Fromanoptimization
perspective,suchaparameterrepresentsadirectfeedbackoftheairfoilfitnesswithrespecttoparticleerosion.
Largevaluesoferosionrateareresponsibleforanearliervisibledamageonthebladeovertimewhencompared
with smaller values. Given the air quality characteristics of a chosen wind farm, a blade section geometry that is
responsibleforhighererosionrateswilloutputlessenergythanamoredamage-resilientbladeovertime. Suchcon-
siderationsmotivatetheinvestigationofoptimumsectiongeometries,underthenovelperspectiveofairborneparticle
erosion. Thegoalofthisstudyistoinvestigatethebladesectiongeometryoptimizationforthecaseofsanderosion
becausesandrepresentsthetypeofparticlethatismostlysusceptibletothebladeaerodynamicflowfield,andhence
thebladesectiongeometry.
Thispaperisdividedintofivesections: thenumericalmethodusedisexplainedinSectionII,thebladeoperating
point,particleaerodynamicsanddamagemodelsareintroducedinSectionIII,whiletheresultsobtainedarediscussed
inSectionIV.Finally,conclusionsareproposedinSectionV.
II. MethodologyandTheoreticalDevelopment
A. ParticleEquationsofMotion
Inordertopredicttheerosiveeffectofsandgrainsonthebladesurface,alagrangianformulationcodewasdeveloped
in-houseandnamedBugFoil.22,25BugFoilintegratesapre-existingparticletrajectorycode33andacustomizedversion
ofXFOIL.34 Thelocalflowfieldvelocitycomponentsareobtainedbyqueryingthebuilt-inpotentialflowroutineof
XFOILfromwhichtheparticletrajectoryandimpactlocationontheairfoilarecomputed.
Insteadyflight,theforcesactingontheparticleareperfectlybalancedandperturbationstosuchforcesareassumed
to be additive to the steady-state forces. For these reasons the equations of motion may be expressed by neglecting
thesteady-stateforcesandmaybewrittenasfunctionsofincrementsonly.35 Inthecurrentstudy,bothraindropsand
hailstonesweretreatedasaerodynamicbodieswhoseonlyassociatedforceistheaerodynamicdragD.
ByapplyingNewton’ssecondlawalongtheparticletrajectoryinbothchordwisexandchord-normalydirections,
thefollowingequationsareobtained18,22,36–38
d2x
P
m =ΣF (1)
P dt2 x
d2y
P
m =ΣF (2)
P dt2 y
By projecting the drag of the particle D in both chordwise x and chord-normal y directions using the relative angle
betweenparticleandflowfieldvelocityα ,theequationsmayberewrittenas
r
d2x
P
m =∆D cosα (3)
P dt2 r
d2y
P
m =∆D sinα (4)
P dt2 r
3of23
AmericanInstituteofAeronauticsandAstronautics
GiventheparticlevelocitycomponentsU andV andgiventhevelocityflowfieldcomponentsU andV atacertain
P P
pointalongthetrajectory,theparticleslipvelocityV canbeexpressedas
s
(cid:113)
V = (U−U )2+(V−V )2 (5)
s P P
whilethetrigonometricfunctionsinEqs.3and4mayassumetheform
U−U V
P rx
cosα = = (6)
r
V V
s s
V−V V
P ry
sinα = = (7)
r
V V
s s
ByexpressingtheparticleaerodynamicdragDasafunctionofdynamicpressureandbysubstitutingforthetrigono-
metricfunctions,theEqs.3and4mayberewrittenas
d2x 1 V
m P = ρV2A C rx (8)
P dt2 2 s P D V
s
d2y 1 V
m P = ρV2A C ry (9)
P dt2 2 s P D V
s
Toscalethisprobleminanon-dimensionalfashion,non-dimensionaltime,space,andmassparameterscanbeintro-
ducedhere
tU
τ = (10)
c
x
P
x = (11)
P
c
y
P
y = (12)
P c
2m
P
AK= (13)
ρ A c
P
NondimensionalizationofEqs.8and9byareferencevelocityU yield
d2x 1
P
= V C V (14)
dτ2 AK r D rx
d2y 1
P = V C V (15)
dτ2 AK r D ry
whichtogetherrepresentasetofsecond-order, nonlineardifferentialequations. Oncetheparticledragcoefficientis
evaluated,thetrajectorycanbecomputedbynumericallysolvingbothxandyequations.
B. SandErosion
Upon impact, sand grains promote a mechanism of surface abrasion. Sand erosion is responsible for an increase in
bladesurfaceroughnessandadecreaseinstructuralstiffness. TheparametererosionrateE,definedastheremoved
mass of the target material divided by the mass of the impacting particle, is a function of particle impact velocity
V andangleatimpactθ,anditismeasuredinunitsof(g/g).19 ThevelocityV isrelatedtoE throughapower-
imp imp
law; whereas, the correlation with impact angle strongly depends on the eroded material properties. Most current
materials used for wind blade coating are polyurethane derivatives30 and show a primarily plastic erosion behavior
with maximum erosion rate at θ = 30 deg.39 Following the approach previously implemented by the authors,22 the
erosionrateforplasticmaterialsisgivenbytheequation20,40–43
E=KVn (16)
imp
whereK andnareconstantsoftheerodedmaterial. ThecorrelationbetweenE andθ isimplicitintheparametersK
andnfittedatvariousimpactanglesandimpactvelocities.Similartopreviousstudies,thesimulationswereperformed
4of23
AmericanInstituteofAeronauticsandAstronautics
byusinglinear-fittederosionconstantsofultrahighmolecularweightpolyethylene(UHMWPE)22 becauseithasthe
bestperformanceofthepolyethylene-basedcoatings,andbecausenormalimpactscausesmallerosionrates.40
Eachsimulationisperformedbyplacingaverticalarrayofsandparticlesfivechordlengthsupstreamoftheairfoil.
Oncethethesimulationisinitiated,thetrajectoryoftheparticlesisevaluatedandimpingementisestimated.Giventhe
angleandvelocityatimpact,E assumesdifferentvaluesintheregionaroundtheleadingedge(LE),withaminimum
associatedwithnear-normalimpact.
III. AerodynamicsofDamagedAirfoilsandAnalysisofSectionGeometries
The design point for wind turbine airfoils is associated with the maximum aerodynamic efficiency (C/C ) . To
l d max
illustrate the significance of blade shape optimization it is of particular interest to predict the performance of blade
sectionsinadamagedconfiguration(i.e.,whentheaerodynamicefficiencyisreduced).Inthissectiontheaerodynamic
performanceofadamagedDU96-W-180arediscussed.
A. ParametricStudyontheLocationofFixedTransition
The erosion rate E curves due to sand on a DU 96-W-180 airfoil (α = 6.0 deg) are shown in Fig. 1. The earliest
damagetoappearonthesurfacecorrespondstothelocationofmaximumerosionrate,whenthebladeisexposedto
aprolongederosion(asdiscussedbytheauthorsinpreviousworks22). Infact,atsuchlocations,onecanassumethe
quantityoferodedmaterialtobethelargest. Therefore,twofixedtransitionlocationsareconsideredontheupperand
lowersurfaces: xu =3.13%andxl =8.39%,respectively. ThesechordwiselocationsareobtainedwithBugFoilfor
tr tr
abladesectionlocatedatr/R=0.75andα =6deg,sinceitrepresentsasuitabledesignpointnear(C/C ) .
l d max
For a more general scenario, the effects on E due to the blade angle of attack, particle diameter, and section
spanwiselocation(throughlocalbladevelocity)areshowninFig.2. Itcanbeseenthatanincreaseinα [Fig.2(a)]
is translated into a shift of the maximum erosion rate toward the leading edge for the upper surface and toward the
trailingedgeforthelowersurface. Ontheotherhand,anincreaseinparticlediameter[Fig.2(b)]isresponsibleforan
increaseinmaximumE onbothairfoilsides. Also,ashiftofthepeaksoccurstowardthetrailingedgeontheupper
surfaceandtowardtheLEonthelowersurface. Lastly,anincreaseinbladevelocity[Fig.2(c)]causesanincreasein
maximumE thatisroughlyproportionaltoV3. Inallcases,however,itcanbenoticedthatthehighestvaluesofE are
∞
onthebladeuppersurface.
The current parametric study shows that it is relevant to investigate the effects of the chordwise position of the
aerodynamictransitionimposedbytheerosionontheairfoilaerodynamicperformance. Therefore,theaerodynamic
performanceoftheDU96-W-180airfoilwasanalyzedwithXFOILbyparameterizingthechordwisepositionofupper
(U)andlower(L)fixedtransitions. TheeffectonC/C duetothechordwisetransitionlocationcanbeseeninFig.3.
l d
Twocurvesareproposed,eachobtainedbymodelingonesurfacewithchordwisevariabletransitionwhileimposinga
fixedtransitionpointduetosanderosionontheoppositesurface(xu =3.13%,andxl =8.39%).Itcanbeseenthatthe
tr tr
locationofaerodynamictransitionhasadetrimentaleffecton(C/C ) assoonasthetransitionpointmovesfromits
l d max
naturallocationtowardtheLE.Themostsignificanteffectontheaerodynamicefficiencyisduetothetransitiononthe
Figure1.ErosionrateEcurvesonaDU96-W-180airfoil,α=6.0deg.Redsegment:uppermaximumerosionrate(Eu );cyansegment:
max
lowermaximumerosionrate(El );bluecircles:particleimpingementpoints.
max
5of23
AmericanInstituteofAeronauticsandAstronautics
(a)
(b) (c)
Figure2.Effectof(a)bladeangleofattackα,(b)sandgraindiameter,and(c)bladevelocityV∞onsanderosionrateEontheDU96-W-180
airfoil.
Figure3.EffectonC/C ofthechordwiselocationoffixedtransitionfortheDU96-W-180.
l d
6of23
AmericanInstituteofAeronauticsandAstronautics
uppersurfacewhereadecreaseofC/C ≈40%isassociatedwithanearlyuppertransition. Thisfigureisrelevantin
l d
showingthemagnitudeoflossinGAEPofanerodedwindturbine. Similarvalueswereobtainedfortheexperimental
investigationofdamagedwindturbineairfoilperformancecarriedoutinpreviousstudiesbySareenetal.44
B. PolarsofaBladeSectionSubjecttoSandErosion
TheaerodynamicpolarsoftheDU96-W-180airfoil(Fig.1)arecomputedbyusingXFOILforcleansurfacecondi-
tions,upperonlyfixedtransition(U)(setbyerosion),loweronlyfixedtransition(L)(setbyerosion),andbothupper
andlowerfixedtransitions(U+L).AllcomputationswerecarriedoutatRe=6.88×106 asitrepresentsareasonable
Reynoldsnumberforabladesectionlocatedatr/R=0.75andc=1.7m.
Figure 4 shows the effects on the aerodynamic performance due to the various combinations of fixed transition.
By assuming that the earliest damage in time would appear on the blade upper surface, an initial large increase in
C is observed with respect to the clean configuration. Although smaller, a further increase inC is observed once
d d
thebladelowersurfacedamagealsoappears. Finally,adragcoefficientapproximatelydoublethecleanC forU+L
d
transitionsisobserved. AlargercontributiontoC oftheupperfixedtransitionmaybeexplainedbyconsideringthe
d
pressure gradients acting on the upper surface and the longer curvilinear path covered by the boundary layer on the
bladeuppersurfacewhencomparedwiththelowersurface. Infact,athickerboundarylayerwouldbeobservedonthe
uppersurfacewhencomparedwiththeboundarylayeronthelowersurface. Thiswouldcontributetotheoverallform
dragofthebladesectionwhichwouldincreasemorefortheuppersurfacecontributionthanforthelowersurface.
TheaerodynamicefficiencyC/C plotinFig.4showsasimilareffecttoFig.3. Itcanbeseenherethatthedrop
l d
inefficiencyfromcleantofixeduppertransition(U)isdrasticandbringsthebladesectiontoa≈40%penalizationin
C/C atα =6deg. Oncethelowerbladetransitionisalsofixed(U+L),afurtherdecreaseinaerodynamicefficiency
l d
isobserved,bringingthebladesectiontoapproximatelyhalftheaerodynamicefficiencyoftheinitialcleancondition.
Finally,byobservingFig.5aninitialdropinC canbeseenforuppertransition,followedbyamodestrecoverywhen
l
U+L-transitionsareset. Suchbehaviormaybeduetotheover-predictioninliftcoefficientofXFOILandshouldbe
verifiedwithhigher-ordermethodssuchasCFD.Itshouldbenotednowthatthechronologicalhistoryofdamageon
thebladesectionisnotcrucialtothepresentanalysis.Infact,identicalconsiderationsmaybedrawnwhenconsidering
anearlylowerdamageandalaterupperdamage.
C. TheEffectofBladeShapeonSandErosion
The scenario depicted in Sections III. A and III. B does not include the relationship of the blade section shape to
sand erosion rate E. In the current section, the role of different geometry features with respect to sand erosion is
investigated.Anefficientwaytocomparetheresultsistofixtheparticlesize,thefreestreamconditions,andtheairfoil
operating point while varying the airfoil geometry. At this point, it is significant to choose a design lift coefficient
C rather than an angle of attack, since it gives a better indication of the blade operating conditions once the airfoil
l
geometryisincludedinthebladedesign.
The analysis that has been carried out in previous studies22,25 involved the Delft University wind turbine airfoil
familyappliedtothedamageproblem.12,45,46Eventhoughthischoiceofairfoilsiscurrentlyrepresentativeofstate-of-
the-artwindturbines,itissomewhatlimitedwhentryingtoinvestigatethegeometricfeaturesthatpotentiallyminimize
damagetobladesections. Infact,theNRELairfoilfamilyoffersawidervarietyofupperandlowersurfacetopology
andcurvaturewhencomparedwiththeDUfamily.47–49 Forsuchreasons,theNRELSfamilyofairfoilsisincluded
here. Finally, because the most prominent effects of particle damage are seen when moving toward the blade tip,
interestisaimedatoptimizingoutboardbladesectiongeometries. Aliftcoefficientof1.0ischosenandtheresultsare
presentedasacomparisonbetweentwopairsofairfoilsasshowninFigs.6,and7. Theerosionpeaksontheairfoil
uppersidearedenotedbyredmarks; whereas,thecyanlinesdenotepeaksofthelowerside. Itshouldbenotedthat
moreairfoilswereanalyzedbytheauthors,butonlyfourareshowninthissectionforbrevity.
A DU96-W-180 and aDU 96-W-212 arecompared in Fig.6 since theyare characterized byvery similar upper
geometries,whereasthemaximumthicknessisduetothedifferentcurvatureofthelowersurface. TheupperE peaks
arevirtuallyidenticalinvalue,shape,andchordwisemaximumlocation. Ontheotherhand,themorebulbousfront
part of the DU 96-W-212 promotes an erosion peak on the lower side that is noticeably farther downstream, when
comparedwiththethinnerDU96-W-180airfoil. Inotherwords,thefullershapeoftheairfoillowersideallowsfor
the erosion peak to move downstream while also becoming flatter. No appreciable effect is detected as far as the
maximumvalueofE onthebladelowerside.
7of23
AmericanInstituteofAeronauticsandAstronautics
Figure4.C,C ,andC/C curvesoftheDU96-W-180forcleananddamagedconditions.
l d l d
Figure5.DragpolaroftheDU96-W-180forcleananddamagedconditions.
8of23
AmericanInstituteofAeronauticsandAstronautics
Figure6.ErosionratecomparisonforaDU96-W-180andaDU96-W-212atCl=1.0,effectofthelowersidegeometry.Uppersideerosion
ratepeaksinred;lowersideerosionratepeaksincyan.
Figure7. ErosionratecomparisonforaNRELS817andaNRELS831atCl=1.0,effectoftheuppersidegeometry. Uppersideerosion
ratepeaksinred;lowersideerosionratepeaksincyan.
9of23
AmericanInstituteofAeronauticsandAstronautics
ThesituationisreversedwhenconsideringtheairfoilsNRELS817andS831inFig.7. Suchairfoilshavesimilar
lowersidegeometriesintheforwardsection,whereastheuppersideismorebulbousfortheNRELS831. Thelower
erosionpeaksarevirtuallylocatedatthesamechordwiselocation,althoughtheshapeofE appearstobeflatterforthe
NREL S817 airfoil. An important difference appears when considering the upper erosion peaks. Again, the bulkier
geometrypromotesanerosionpeakfartherdownstreamoftheLE,whencomparedwiththethinnersection. Also,a
slightdropinthemaximumvalueofE isseenwhenconsideringtheNRELS831.
D. LessonsLearned
Theanalysisoftheprevioussectionshighlightstheroleofthebladegeometrywithrespecttoerosion. Itcanbecon-
cludedthatthechordwiselocationofmaximumE istranslatedintoappreciableeffectsontheaerodynamicbehavior
ofthebladesectiononceerosionisinitiated. Ontheotherhand,themaximumvalueofE isalsodrivenbytheblade
sectionshape. Inotherwords,theshapeofthecleanbladecandictatewhenandwheretheearliestdamagewouldap-
pearonthesurface;thebladeshapealsodictatestheperformanceinroughconditionsduetothechordwisetransition
locationontheupperandlowersurfaces. Suchanobservationisanimportantfeedbackonbladegeometrysubjectto
sanderosionwithrespecttominimizingthelossofaerodynamicperformanceoncetheairfoilisdamaged. Ingeneral,
thebladedesignerwouldfavorgeometriesthatpromotethelowesterosionpeakslocatedasdownstreamaspossible
fromtheLEtherebyallowingforbetteraerodynamicperformanceinroughconditions. Fromthecomparisonofthe
airfoilsitcanbeconcludedthat:
• An important geometric feature that pushes the maximum-E location downstream is the LE slope. In partic-
ular, it appears that a more bulbous upper side allows for the maximum E to appear more downstream, when
comparedwiththinnerLEgeometries.
• Asimilarobservationcouldbemadeforthebladelowerside. However, itwasobservedthatthecurvatureof
theblademayplayanimportantroleonboththeshapeoftheerosionratecurveandthemaximumlocationof
E (seeFig.6).
• The sensitivity of maximum E to variations in geometry is smaller when compared with the sensitivity of the
chordwise location of maximum E. However, even small changes in maximum E translate into longer blade
life, extending the clean, smooth conditions in which the turbine operate, yielding relatively higher values of
GAEPovertime.
E. FigureofMerit
The considerations of Section III. D pose a new problem that is quantifying the fitness of an airfoil subject to sand
erosion. Generally,itwouldbesignificanttocombineboththemaximumvalueofE andthecorrespondingchordwise
location to obtain a numerical insight on the airfoil erosion performance. In particular, a designer would want to
penalizehighvaluesoferosionrate,whileobtainingtransitionlocationstobeasdownstreamaspossibleontheblade
surface. Forthesereasons,ifsuchadesignerwouldwanttocreatethebesterosion-resilientairfoilofthosepreviously
considered, he or she would need to combine the upper airfoil geometry of the NREL S831 [Fig. 7] with the lower
geometryoftheDU96-W-212[Fig.6].
The proposed mathematical expression to obtain the fitness of an airfoil subject to sand erosion is the following
figureofmerit(FOM),thatis
FOM= 1 (1.1+xu)ωu+ 1 (cid:16)1+xl (cid:17)ωl (17)
Eu tr El tr
max max
whereω andω aretheweightsassignedtotheupperandlowersurface,respectively,inordertopenalizelargevalues
u l
of the chordwise location of transition x . Such aspect is tied with the analysis of Section III. A and represents a
tr
powerfulwaytofavoronesideoftheairfoilgeometrywithrespecttotheother. Becausetheuppersidehasshowna
moreimportanteffectontheairfoilperformanceindamagedconditionscomparedwiththelowerside,thetwochosen
weightsareω =5andω =1.5.
u l
ToshowthevalueofusingFOM,acomparisonofairfoilsisproposed,withfixedC =1.0. Theairfoilsdiscussed
l
in Section III. C, along with NREL S804, S832, S813, and S810, are ranked based on the erosion performance
10of23
AmericanInstituteofAeronauticsandAstronautics