Surrogate model of complex non-linear data for preliminary nacelle design

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Aerospace

Science

and

Technology

www.elsevier.com/locate/aescte

Surrogate

model

of

complex

non-linear

data

for

preliminary

nacelle

design

Alexander Heidebrecht

∗

,

David

G. MacManus

CranfieldUniversity,MK430ALCranfield,Bedfordshire,UK

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received11February2018

Receivedinrevisedform4August2018 Accepted18August2018

Availableonline3October2018 Keywords:

Nacelledragresponse Surfacemodel Surrogatemodel Preliminarydesign

Mostresponsesurfacemethodstypicallyworkonisotropicallysampleddatatopredictasinglevariable andfittedwiththeaimofminimizingoverallerror.Thisstudydevelopsametamodelforapplicationin preliminary designofaircraftengine nacelleswhichis fittedtofull-factorialdata ontwooftheeight independent variables,and aLatinhypercube sampling onthe othersix.The specificset of accuracy requirements for the key nacelle aerodynamic performance metrics demand faithful reproduction of parts ofthedata toallow accuratepredictionofgradients ofthedependent variable,butpermitless accuracy onotherparts. Themodel is used to predictnot just theindependent variable butalso its derivatives,andtheMachnumber,anindependentvariable,atwhichacertainconditionismet.Asimple Gaussian process model is shown to be unsuitable for this task. The new response surface method meets the requirements bynormalizing theinput data to exploit self-similaritiesin the data.It then decomposestheinputdata tointerpolateorthogonalaerodynamicpropertiesofnacellesindependently ofeachother,andusesasetoffiltersandtransformationstofocusaccuracyonpredictionsatrelevant operatingconditions.Thenewmethodmeetsalltherequirementsandpresentsamarkedimprovement overpublishedpreliminarynacelledesignmethods.

©2018TheAuthors.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY license(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Responsesurfacemodels(RSM)areinterestinginthedesignof engineeringsystems, as they promise not justnear-instant eval-uation ofcomplicateddesigns butalso to maximize theuse that canbe made ofexisting data.Forthis reason,they havebecome frequently used in various engineering studies, both to augment andaccelerateoptimisationprocesses[1] andassubstitutesforthe higher-fidelity methods used to generate the training data, par-ticularly in preliminarydesign [2]. Since the development of the originalKriging method[3], severalvariants were developed,and improvementshavebeenfound,leadingtoageneralizationinthe form of Gaussian processes. Standardized modelling approaches havebeendevelopedforarangeofapplications[4].However, sev-eral shortcomings have also been noted [5]. These include the simplicityofacademicproblemsusedtodemonstrateRSM perfor-manceas well as the assumption of univariate output. Although themethodsarecontinuouslybeingdeveloped[6],therearemany real-world applications to which the current best practices for Kriging models cannot be directly applied. This may be because

*

Correspondingauthor.

E-mailaddress:a.heidebrecht@cranfield.ac.uk(A. Heidebrecht).

data cannot be easily sampled in the recommended way or be-causethedatacharacteristicschangeveryquicklyfromcontinuous to highly non-linear. In some cases the metric of interest is not the predictedvariable itself butits derivative withrespect to an independent variable, or the value of the independent variables at which thedependent variableexhibits a given feature. At the sametime,preliminarydesignapplicationspermitcomparablylow accuracy on absolute performance figures as long as the effects of design changes are modelled correctly. This allows the use of methods withveryselectiveaccuracy, wheredataof lowinterest canbesimplifiedtoalargeextent.

Thisstudyinvestigatesapreliminarydesignproblemwhereall of the above difficulties arise and develops an RSM to predict multiple performance criteria. Sincethe studywas limitedto es-tablished methodswith readily-availableimplementations, it was chosen to use a combination of several Gaussian process mod-els. Due to the resources needed to produce training data, and specific researchinterests, themodellingtechniques developedin thisstudyareusedexclusivelyontheproblemofaero-engine na-celle design.The resultingRSMisintendedforuseinpreliminary nacelle design studies, and in trade-off studies investigating the wider field of engine design and integration. However, it is be-lieved that the generalapproach to establishing an interpolation model,orsome of theRSMcomponents maybe usefulforother

https://doi.org/10.1016/j.ast.2018.08.020

Nomenclature

Abbreviations

CFD ComputationalFluidDynamics CST Class-ShapeTransformation DoE DesignofExperiments GCI GridConvergenceIndex LHS LatinHypercubeSampling OK OrdinaryKriging

RSM ResponseSurfaceMethod Symbols

A Area

cD nacelledragcoefficient,relatedtohighlightarea c_D normalizeddragcoefficient

c_D meannormalizeddragcoefficient

cD,re f referencedrag

g1

...g

6 non-dimensionalnacelledesignvariables

fmax non-dimensionalpositionofmaximumnacelleradius

fnx coefficientinnuggetequation lnac nacellelength

lnx limitinnuggetequation

M flightMachnumber

M_T normalized,transformedMachnumber

MD R dragriseMachnumber Mre f referenceMachnumber MFCR mass-flowcaptureratio

˙

mintake intakemass-flow

n nugget

r radialcoordinate

rhi nacellehighlightradius

ri f initial forebody radius: radius ofcurvature atnacelle

highlight

rmax maximumnacelleradius

rT E nacelletrailingedgeradius

v flowvelocity

x independentvariable

y dependentvariable

z dependentvariable

βnac

boat-tailangle

γ

isentropicexponent

ρ

density

σ

standarddeviation Modifiers

Xcruise cruisecondition

XE O C end-of-cruisecondition

Xhi nacellehighlight

Xi index

Xmax largestvalue

Xmin smallestvalue

XT transformedquantity

X_∞ stateatupstreamfar-fieldposition

X normalizedquantity

x

difference

Qx quotient

problems,asit deals withchallengeswhich are frequentlyfound inengineering.

1.1. Aerodynamicnacelledesign

The problemofthe preliminarydesign of turbofannacellesis becomingincreasinglyrelevant,duetotheincreaseinfandiameter whichistypicallyassociatedwithlow-specific-thrustaero-engines withhighbypassratios.Thisincrease isaconsequenceofthe de-mandforquieterandmorefuel-efficientenginesandthefactthat both goalsrequirean increase in propulsiveefficiency,leading to reduced specific thrust [7]. This generally means an increase of the outer engine radius and thus the surface of the nacelle.The increased weight and drag from the nacelle counteracts the ef-ficiency gained by reducing specific thrust [8,9]. This eventually outweighs the propulsive efficiencybenefits [10]. Forthis reason it is importantto assess theaerodynamic propertiesof potential nacellesearlyintheenginedesignprocess.

Inthiscontext,thepreciseachievabledrag,althoughimportant, is not the only focus of attention. A nacelle must also be able to functionat arange ofoperating conditionswithout exhibiting adverse behaviour.Geometrical constraints for nacelledesign are imposed by the intake andnozzle design and the need to pro-vide sufficientvolumeto accommodateauxiliary unitsandthrust reversers.Thetaskforanacelledesignerinthisscenarioisto de-terminetheminimal overall dimensionsrequiredfora nacelleto fulfil all mechanical and aerodynamic requirements, andto esti-matetheresultingdragofsuchadesign[11].

1.2. Performancemetricsofnacelles

Fig.1ashowsthedragofanexamplenacelledesignasa func-tionoftheaerodynamicoperatingconditions(M,MFCR).Thedrag map is not rectangular since the maximum achievable MFCR is

limitedby theintakecapacity,whichdependsontheflight Mach numberandtheintakedesign(seeSection1.1).

Nacelle drag coefficient varies only little at low flight Mach numbers andlargeMFCR, andstays ata lowbaseline level, com-pared to other operating conditions. Towards both large flight Mach numbers andsmall MFCR, the drag increases, due to com-pressibilityeffects andtheaccelerationaroundtheprofileleading edge,respectively.Fromanacelledesignperspective,thisdrag in-creaseshouldnottakeplaceattheconditionsforwhichanacelle is being designed, both to avoidhigh nacelle drag itself and be-cause a nacelle with a very large shock or separated flow may also influence the flow over the wingand causeadditional drag orlossoflift.Fig.1bshowsso-calleddrag-risecurveswhich high-light how the sharpincrease to nacelle drag as a function of M

is influencedby MFCR.Formanydesigns, wave drag canbecome more than one order ofmagnitude larger than the other nacelle dragcomponents.Fig.1cshowsthattheeffectofMFCRonnacelle dragismoregradualthanthewavedragrisebutisalsonon-linear. During a typical flight, the aircraftwill take-off at M

≈

0

.

25,

MFCR

≈

2,then accelerate andclimb atcloseto maximum MFCR

until it has reached cruise, usually at M

>

0

.

8. For the purpose of this study, it is assumed that there is a margin of 0.02 be-tween the cruise Mach number(Mcruise) and thedrag-rise Mach

number(MD R),atwhichwavedragstartstorisesharply(Eq.(2)). There isno universaldefinitionfor MD R,andthisstudyuses the gradient-based criterion definedin Eq. (1). Not all response sur-facesusedinthisstudyprovideastraightforwardwaytocompute derivatives. In order to keep a consistent approach, all drag gra-dients from response surfaces were evaluated using finite differ-ences,with

M

=

10−4_{.Onceatcruisecondition,theengines are}

throttledbackslightlytosustainaltitudeandvelocity.WhileMFCR

canvarysignificantlybetweendifferentengines,itusuallystaysat orabove0.7atthestartofcruise.Forthisstudy,arepresentative

Fig. 1.(a)Dragmapofanexamplenacelle.(b)cD asafunctionofM(“drag-risecurves”)atconstantMFCR.(c)cD asafunctionofMFCR(“spillagecurves”)atconstantM. Dragcoefficientsusehighlightareaasreferencearea.

mass-flowcaptureratioof0.75wasassumedforthecruise condi-tion(Eq. (2)).As theaircraftburnsfuelduring cruise,it becomes lighterandrequireslessthrust towardstheendofthecruise seg-ment. Depending on nacelle design and the mission, MFCR may be reduced to 0.6at end-of-cruise. During descent, MFCR is fur-therdecreased to 0.6 orbelow. Values of 0.5or lessare usually only reached in situations like engine failure. In such cases, the additionaldragfromthenacelle causesayawingmoment,which meansthatthereisalimittohowmuchadditionaldragsucha re-ductioninMFCRmaycause[10].Theresultingrequirementsfora preliminarydesignmethodareaccuratepredictionofnacelledrag atMFCR

>

0

.

6,withreducedaccuracyacceptablebelowthismark. Thelower bound forthe regarded MFCR rangewas chosen to be 0.4 MD R

=

M

dc D dM=0.1 (1) MFCRcruise

=

0

.

75

,

Mcruise

=

MD R

(

MFCRcruise

)

−

0

.

02 (2) 1.3.DatacharacteristicsandchallengesforRSM

Aresponsesurfacemethodmodelsavariableasafunctionofa setofindependentvariables.Often,an implicitassumptionisthat theseindependentvariablesarecreatedequal,i.e.thatanisotropic samplingmethodcan be used [5], such asLatin hypercube sam-pling (LHS)[12,13]. Another frequentimplicit assumption is that predictionsneedtobeequallyaccurateforallcombinationsof in-putvariables,andthatnoisecharacteristics aresimilar acrossthe sampled region. Lastly, most response surfaces are only used to predictthevalue ofonedependentvariable[5],whereastheRSM athandneeds to alsoprovide valuesofindependent variablesat whichacertaincondition isfulfilled,andmultiplepiecesof infor-mationderivedfromthedependentvariable.

For the response surface model developed in this study, the assumptions outlined above are violated. The following Sec-tions1.3.1–1.3.3describethesechallengesinmoredetail.

1.3.1. Datasampling

Theresponsesurfaceregardedinthisworkmodelsnacelledrag coefficient,cD,asafunctionofasetofsixgeometricaldesign vari-ables(g1 through g6),aswellasthetwovariableswhichdescribe

theoperatingconditions,M andMFCR.Fromthisdata,other per-formancemetricssuchasMD R arethenderived.

The generally recommended sampling plan for RSMs is Latin hypercubesampling(LHS)[4,13].However,thiswouldrequirethat for each nacelle design, the drag is sampled at only one single operatingcondition.Onceanacellegeometryisgenerated,amesh createdandaconvergedsolutionisobtained,itismuchmore com-putationallyefficient to generatemultiple solutions forthe same nacelledesignatdifferentoperatingconditionsthanitistorepeat

thefullCFDprocessforeachdatapoint.Amuchlargeramountof datacanbegeneratedforagivencomputationalbudgetbydensely samplingnacelle drag along the M andMFCR axes andsampling morecoarselyingeometricaldesignspace.Anotherreasonto sam-ple anisotropically is the need to compute gradients of drag as a function of flight Mach number, which is required to evaluate

MD R (Eq.(1)).Sincethedrag gradientchangesrapidlywithin the

transonicrange(0

.

75

<

M

<

1

.

0),intervalsbetweensamplesneed in this range need to be on the order of 0.01. Similarly dense samplingon theother independentvariables wouldbe computa-tionallyprohibitive.Thismeansthatnotallindependentvariables canbetreatedequallybytheRSM,andthatthetwovariables de-terminingtheoperatingconditionneedtousesamplingplansthan thesixgeometricaldesignvariables.

Sincethespaceofvalidoperatingconditionsisnotrectangular (Fig. 1), the samplingplan in(M, MFCR) cannot be asimple full factorial or a LHS. Either a non-rectangular sampling plan needs tobe used,oranaffinetransformationmustbe appliedtomapa rectangularsamplingplanontothespaceofvalidoperating condi-tionsinadragmap.

1.3.2. Varyingdatacharacteristicsandaccuracyrequirements

Fig.2showshownacelledragcharacteristicscanchangerapidly with operating conditions, andacross different designs. Accurate drag prediction is mostly relevant atcruise Mach numberat the cruiseconditionandMFCRsdownto0.6(Section1.2,Eq.(2)).The cruiseMachnumberisbelowdragrisebydefinition(Eq.(2)),and additionaldragincurredwhilereducing MFCRfromthiscondition islowforallacceptablenacelledesigns.ForNacelledesignswhich donot meettheseacceptabilityrequirements,an accurate perfor-mancepredictionisnot required.Thismeansthat, effectively,the operatingconditionsforwhichaccuratepredictionsareneededare always withinthepartofthedragmapwithonlysmallvariations in drag. Outside of thisrange, drag prediction is only important asit needs to beknown whethera design meetsthe acceptabil-ity requirements ornot, butnot the exact margin by which the requirementsaremissed.

Thedrag-risecriterion(Eq.(1))alsoresultsinanadditional re-quirementontheRSM,topredicttherangeofoperatingconditions forwhich aparticular nacelledesign provides sufficientdrag-rise margin.Bytestingthisrangeagainstthemissionprofileofa partic-ularaircraft,thedesignercandeterminetheviablerangeofnacelle designs tochoosefrom. The resultingrequirementontheRSM is accuratepredictionofMD R,asafunctionofMFCRandthedesign variables. Since MD R dependson drag gradients, thisrequiresan

especially consistentpredictionof cD up to drag-riseMach num-ber,butnotatM

>

MD R.

Therequiredaccuracyvariesnotjustwithoperatingconditions butalsoacrossdifferentregionsinthegeometricdesignspace.Ifa designhasparticularlypooraerodynamiccharacteristics,itis

suffi-Fig. 2.Dragmapsofseveralexamplenacellesdesignswhichfeature(a)lowoveralldrag,moderatespillageandgoodMD R,(b)highdrag,verylowMD R,(c)extremespillage.

cienttoknowthatathresholdhasbeencrossed.Forexample,the veryabruptdragincreaseatlowMFCRinFig.2cdoesnotneedto berepresentedaccurately,aslongasitisknownthatspillagedrag ispastthelimitofacceptability.

Training a response surface modelto model such highly non-linearpartsoftheinputdatameanstuningthemodelparameters to represent the very rapid and erratic changes to drag seen in Fig.2catMFCR

<

0

.

6,ortheconsiderabledrag gradientsatflight Machnumbers abovedrag rise.Sucha modelwouldnotbe opti-mallysuitedtorepresentthemuchsmootherdragmapsof desir-ablenacelles(e.g.Fig.2a),andwouldbeexpectedtohavereduced accuracy in more pertinent regions of the drag map (M

<

MD R,

MFCR

>

0

.

6).Onewaytoaddresssuchaproblemistoreducethe sampledspaceto exclude regions ofundesirable data [14]. How-ever, thisis not an option in the current studysince the extent ofthe problematic regionis not known a priori, since the range ofaccuratepredictionsneedsto includethetransitionregions to-ward undesirable data,andbecause even forundesirable designs or unrealistic operating conditions, an indication of drag is still needed.Asuitable RSMmustthusprioritizethemodellingof de-sirablefeatures,whilestillprovidingindicatorswhenperformance isnotacceptable.

1.3.3. Datashiftonindependentvariables

Oneof themain performance figures fornacelles isthe drag-rise Mach number (MD R), as it determines the maximum flight

Machnumberatwhichthenacellecanoperatewithanacceptable drag. This posesa challenge:Fundamentally, MD R isnot the de-pendentvariablepredictedbytheRSM,butratherthevalueofan independentvariableatwhichthecharacteristicsofthedependent variable change. This very distinctive feature of every drag map shiftsalongtheMaxisasthedesignchanges,similartotheeffect showninFig.1b,whereMD R changesasafunctionofMFCR.

The dense sampling on the M axis required to correctly de-termine MD R cannot be feasibly usedon theother axes,butthe resulting RSM still needs to provide the necessary level of de-tailtodetermine MD R atpoints inthe designspaceatwhichno

input datahasbeensampled.This cannot beexpectedof a stan-dard interpolation method which only regards amplitude of the independent variable(cD). As an example,a hypotheticalKriging methodtrained on thetwo drag-risecurvesat MFCR

=

0

.

65 and

MFCR

=

0

.

75 inFig.1b,isregarded.Althoughthemaindifference betweenboth input curves is a shift of about0.05 along the M

axis,aninterpolationtoMFCR

=

0

.

7 wouldbeexpectedtoproduce ablendbetweentheinputcurves,wheredragbeginstoriseearly butmoregraduallythanineitheroftheinputcurves.

Tosolvethisproblem,asuitableRSMmustmodelnotjustthe nacelle drag accurately butalso has to includesome measureof the shift of the drag-rise feature along the Mach number axis which occursasa result ofchanges to MFCR or the design vari-ables.Anaïveinterpolationmethodwhichsimplymodelsdrag as

Fig. 3.The parameters used to specify the nacelle geometry.

afunctionofallindependentvariablescannotbeexpectedto per-formwellinthisregard.

1.4. Inputdatageneration

The data used in this study was obtained with the process foraxisymmetricCFDsimulationsfornacellesdescribedby Heide-brechtetal.[15].Theprocessisbuiltaroundaparametricnacelle geometry(Fig.3).ThenacelleisconstructedfromaCSTcurve[16], constrainedby ananalytical approach[17] toconformto asetof six intuitive design variables.The geometrical set-up and the di-mensionaldesignvariablesare showninFig.3,andaresimilarto the set-up used inother CFD simulations withthe aimof deter-mining nacelledrag [18]. Thedesign variables aremapped tosix non-dimensional degreesoffreedomtodefine a non-dimensional design space (Table 1). The non-dimensional variables define the nacelleshaperelativetothehighlightradius,rhi.rhi waskept con-stantforalldesignsintheinputdataset,asitonlyaffectsthescale andhasthereforeonlyasmalleffectonaerodynamiccoefficients, intheformofReynoldsnumbereffects.

The CFDsimulations were conductedwitha compressible im-plicit flowsolver [19],withthe assumptionofa perfectgaswith a constant isentropic exponent of

γ

=

1

.

4. They employed the

k–

ω

SST turbulence model. The domain inlet boundary condi-tion was set to the farfield total pressure, to generate an efflux which matches the farfield velocity. Thiswas done togenerate a generic exhaust conditionindependent ofanyspecific engine de-sign, andto minimize post-exit forces, which are not present in ideal pressure-matchedexhausts [20]. The mesh consistedof ap-proximately 39,000 cells, and a mesh sensitivity study found a gridconvergenceindex(GCI)[21] of1%onnacelledrag,basedon mesheswithhalvedanddoubledresolution (11,000and155,000, respectively).Adomainsensitivitystudyuseddomainsbetween50 and90nacelleradii,andfoundtheinfluenceofdomainsizeon na-celledrag tobeontheorderof0.01%,forthechosendomainsize of 80 nacelleradii. Nacelle drag was extractedasmodified stan-darddrag [20],usingthemethoddevelopedbyChristie [9].Fig.4

illustrates a typicalconvergence history,amesh andaflow solu-tion, forthe samedesignused toproducethe datainFig. 1.The

Table 1

Dimensionalandnon-dimensionaldesignvariablesfortheparametricnacellegeometry,andthedesignspacelimits.

Variable Description Non-dimensional min max

rmax Maximum radius g1=(rmax_l−rhi)

nac 0.017 0.126

lnac Nacelle length g2=l_rnac_hi 2.6 4.5

fmax Location of maximum radius g3=fmax 0.2 0.5

ri f Initial forebody radius g4=fi f=

ri f·fmaxlnac (rmax−rhi)2

0.6 1.5

fte Trailing edge radius g5=rte_l−rhi

nac −0.05 0.035

βnac Boat-tail angle g6=βnac/◦ 9.5 14.5

Fig. 4.DatafromtheCFDsimulationforarepresentativenacelledesign.(a)ConvergencehistoryforoneMachnumbersweep,atmaximumMFCR;(b)mesh;(c)flowfieldat M=0.845,MFCR=0.78.

CFDapproachwasalsopreviouslyvalidatedagainstmeasurements [22].Fig.4showsanexampleofconvergence,meshandflow so-lutionforarepresentativenacelledesign.

2. Methodology

Although it is computationally possible to generate tens of thousandsofCFDresultsforthesimpleaxisymmetric CFDmodel employed,therequirementtoevaluatemanyoperatingconditions foreach nacelle design means that the number of designs sam-pled in the 6-dimensional design space is much lower than the total numberof samplednacelle drags. Thismay hinder the use ofmethodswhichmightotherwisebeusefulforlargenumbersof samples,anditfavoursanapproachwhichtreatsthedesignspace separately fromthe2-dimensionalspace whichdefinesthe oper-atingconditions.

Consequently,theresponsesurfacemodellingproblemwas de-composedintoseveralcomponents.First,aninterpolationmethod forindividual drag maps is established for the full-factorial data generatedforeachinputdesign.Thisisusedtoextracttwoscaling referencesfromeach ofthedragmapsintheinputdataset.These scalingreferencestothetwomainoutputsofthefullmodel,drag at subcritical Mach numbers, and the onset of wave drag. They

help exploit the self-similarities between drag maps of different designs, andallowthe datato benormalized insuch awaythat themainfeatures ofdragmapsaremapped tothesame normal-izedinputvariablesforalldesigns.TheremainingpartoftheRSM process uses normalized variables, and it models the individual differencesintheshapesofdrag maps,relativetotheir common-alities,asafunctionofthedesignvariables.

2.1. Analysisandinterpolationforsingledesign

The following sections describe how the data obtained for a single design is sampled, how a drag map RSM is created, and how the scaling references are determined and used to non-dimensionalizethedragmapdata.

2.1.1. Datasamplingandnon-dimensionalisation

The aimofthesamplingmethod fordragwithin the spaceof operatingconditions(M andMFCR) istoprovide datasuitable to subsequentlyestablishan accurateinterpolation modelforadrag map. It needs to provide sufficient samplingdensity where drag changessharply,whilenotexpendingtoomanysamplingpointsin areasofthedragmap wheredragchangesvery little.Particularly aroundtheonsetofwavedrag,thesamplingplanneedstoprovide

Fig. 5.(a)Dragmapofanexamplenacelle.Symbolsindicatesamplelocations.(b)cDasafunctionofMatforarangeoffixedMFCR.(c)cDasafunctionofMFCRatdifferent

M.(d),(e), (f)ThesamedataasfunctionsofMTandMFCRT.

sufficientdatatoaccuratelycomputethedraggradientinformation requiredtofindMD R (Eq.(1)).

The maximum possible MFCR at a given Mach number,

MFCRchokecanbecomputedforisentropicflow(Eq.(3)).Becausein

realflowstheintakeboundarylayereffectivelyreducestheintake throatarea, andbecausea flow very closetochoking maycause divergenceinaCFDsolver,themaximumvaluewas setto95% of theisentropicchokingvalue.The lowestsampledMFCRwas cho-sen as 0.4(Eq. (4)), to allow prediction for values of MFCR that maybe reachedinengine-outscenarios(Section1.2).Ratherthan distributingsamplingpointsevenlywithintheinterval,itwas de-cided touse a mapping whichis linear with

√

MFCR−1,mapped to the interval

[

0

;

1

]

(Eq. (5)). The inverted square rootfunction increasesthesamplingdensitytowardslower MFCR,wherelarger drag gradients are expected, and the linear mapping allows the dragmapinterpolationtoworkonaunitsquare.

MFCRchoke

=

1

+

0

.

2M2 1

.

2

3 1 C R

·

M (3)

MFCRmin

=

0

.

4

;

MFCRmax

=

0

.

95MFCRchoke (4)

MFCRT

=

√

MFCR−1

−

√

MFCRmax−1

√

MFCRmin−1

−

√

MFCRmax−1 (5)

ThesampledMach numberintervalwas chosentobe0.2–0.95 (Eq.(6)).Sinceitisknownthataerodynamiceffectswhichare de-pendentonMachnumbermainlyappearathighMachnumbers,a samplingdistributionwaschosenwhichtakesthisintoaccount.In thiswork, thePrandtl–Glauert factor,

√

1

−

M2_{, whichis used in}

thePrandtl–Glauerttransformation[23],isusedtolinearizesome ofthecompressibilityeffects.Thisbiasesthesamplingdensity to-wards larger Mach numbers. Like MFCRT, the transformed Mach numberislinearlymappedtotheinterval

[

0

;

1

]

(Eq.(7)).

Mmin

=

0

.

2

;

Mmax

=

0

.

95 (6) MT

=

√

1

−

M2

−

₁

−

_M2 max

1

−

M2_min

−

1

−

M2max (7)

WithinthesquarecreatedbyMT andMFCRT,alinearfull fac-torialsamplingwasgenerated.Sincethenacelledragisinfluenced much more by Mach number than by MFCR, the sampling plan comprises28samplesalongtheMT axisand7samplesalongthe

MFCRT axis.Fig. 5shows thesamplingplan asa function of the original (M and MFCR) and transformed (MT and MFCRT) oper-ating conditions. It can be seen that the transformation success-fullybiasesthesamplingtowardstheconditionswherethelargest changes in drag occur, andreduce some of thenon-linearities in thedata.Forexample,linesofdragatconstantMachnumbershow lesscurvaturewhenplottedasafunctionofMFCRT (Fig.5c,f).The

drag-risecurves asafunction of MT show a muchmoregradual onset ofwave drag than thesame curve plottedagainst M,with thesegmentsoneitherendofthecurveapproachinglineartrends (Fig.5b,e).

2.1.2. Interpolationofdragmaps

To interpolate within a drag map, a Gaussian process model [24] isgenerated.Thisusestheimplementationinscikit-learn[25]. The implementationoffers theabilityto specifyacustom regres-sionmodel,aswellasalocalnuggetvariabletoexpressthe mea-sureofconfidenceintheaccuracyofagivenvalue.Thedefinition forthenuggetisshowninEq. (8),where

σ

i isthestandard

devi-ationontheinputvalue atindexi.Thebasenuggetvaluechosen forallinputdatawasnbase

=

4

·

10−6,whichcorrespondstoamean

predictionerrorof0.2%intheCFDresults.

n

=

σ

i yi

2 (8)

Itwas foundthatattemptingtomodelallofthepotentially ir-regular inputdata(e.g.Fig.2c)withthisaccuracycancorruptthe RSM,asit leadsto alow estimatedcorrelation betweenthedata points, which results in oscillations throughout the entire map.

Fig. 6.Fourexampledrag-risecurves,sampledatmaximumMFCR.(a)Rawinputdata,cDagainstM;(b)cD againstMT;(c)transformedandnormalizedtoc,re f andMre f,

c_DasfunctionofM_T.

Therefore,thenugget valuesforsome of theirregularinput data were increasedto reduce their impact on themodel parameters. Suchan increase can be justifiedfor datafromparts ofthedrag map where the respective design would not conceivably be re-quiredtoperformwell.Thisisthecasefordataabovedragriseor datapoints withveryhighspillagedrag. Suchdatadoesnotneed tobeaccuratelyreproducedbytheRSM,aslongasitispossibleto provideanindication thatdragisunacceptablyhigh. Evenat con-ditionsbelowdrag-rise,small-scaleirregularitiesmaybesmoothed out.Inthe contextofpreliminary design,thiscorresponds tothe assumptionthatitispossibletoremove suchsmall-scalefeatures fromthedragmapatalaterstageinthedesignprocess.

To deal with non-smooth data, the nugget was increased by twoadditionaltermswhichdependonthesmoothnessofthedrag map,asindicatedbythesecond-orderfinitedifferencesinthe in-putdata, δ2cD δM2 T and δ2cD δ2_MFCR2 T

.Forbothindicators,acceptabilitylimits weredeterminedby analysingthedragmapsfornacelles with ac-ceptableperformance.Shouldtheindicatorfalloutsidetheselimits fora givendatapoint,thenuggetforthisdatapointisincreased proportionallytothesquare ofthemarginbywhicheach limitis violated.

AGaussian process model is then generated to predict cD as

a functionof MT and MFCRT.It uses a squaredexponential

cor-relationmodel[24] andemploysa customregressionfunction,to makeuse ofexisting knowledge aboutthegeneralshape of drag maps.Thecustomregressionmodelconsistsofsecond-order poly-nomialsforboth MT andMFCRT,inadditiontoexponentialterms dependentonMT,aswellasallcombinationsofinteractionterms betweenthetwoinputvariables.Theexponentialtermsallowthe modelto account fora quickly increasing drag gradient with in-creasingMachnumbers.

2.1.3. Scalingreferencesandnormalisation

Toprepare the data forinterpolation between the drag maps ofdifferentnacelledesigns,twoscalingreferencesaredetermined whicharethen usedtonormalizethedata andto makeiteasier toexploitsomefundamentalself-similaritiesbetweenaerodynamic propertiesofdifferentnacelledesigns. The firstscaling reference,

cD,re f waschosen tobethenacelledrag atM

=

0

.

6,MFCR

=

0

.

75

(Eq. (9)). This is representative of the level of the typical drag plateau at low Mach numbers and high MFCR. cD,re f also

indi-cates the level of drag at which accuracy is most important, as thereshould not belarge changes indrag betweenthereference operatingconditions(Eq.(9)),andthecruisecondition(Eq.(2)).

Thesecond scalingreference, Mre f,isdefinedusingthe gradi-entcriterioninEq. (10). Thisusesa verysimilardefinitionasthe drag-riseMach number(Eq. (1)) butusinga larger thresholdfor thedrag gradient.The reasonfor thisis that Mre f can be

deter-minedwithgreateraccuracythanMD R becauseanyinaccuracyin

determiningthedraggradienthasasmallerinfluenceontheresult ifthegradientislarger.

cD,re f

=

cD

(

M

=

0

.

6

,

MFCR

=

0

.

75

)

(9) Mre f

=

M

dc D(MFCRmax) dM =0.5 (10)

Eachdragmapisthennormalizedwithrespecttothetwo scal-ing references.To do this, two newvariables are introduced, the normalized drag, c_D (Eq. (11)), and the normalized transformed Mach number, M_T (Eq. (12)). This normalization maps both the reference drag (cD,re f) andthe referenceMach number (Mre f,T)

tonominalvaluesof1.The differentstagesofthe transformation fromMandcD toM_T andc_D areshownforfoursampledrag-rise curvesinFig.6.Ascanbeseen,manyofthesystematicdifferences betweenthe curvesare removed by the normalization. Most im-portantly,thebulkofthewavedragincreaseappearsatthesame position on the M_T axis for every design, andthe slopes of the curvesatM_T

>

1 arealmostequal.Themainremainingdifferences betweenthedrag-risecurvesareat0

.

8

<

M_T

<

1

.

1 (Fig.6c).These differencesareofmuch smalleramplitude thanintheinput data (Fig.6a)andconcerntheshapeofthecurvesduring theonsetof wavedragrise.

c_D

=

cD

−

cD,re f

+

1 (11)

M_T

=

1

−

MT 1

−

Mre f,T

(12)

This normalisation step allows the subsequent process to use separatemodelsforthedifferentaerodynamicpropertiesofnacelle designs,representedbythescalingreferences(c,re f andMre f)and

thenormalizedinputdata(c_D asafunctionofM_T,MFCRT andthe

designvariables).

2.2. Interpolationbetweenmultipledesigns 2.2.1. Samplingindesignspace

Topopulatethedesignspace,aLatinhypercubesampling(LHS) [12,26], was generated, which covers the entire six-dimensional designspace(g1–g6,Table1). Somecombinationsofdesign

vari-ables lead to nacelle geometries whichare not fully convex, and such designs were rejected since they are expected to have un-acceptable drag characteristics. This process generated 235 valid nacelledesignsfromaLHSofover500samples.Afterconstructing an initial RSM,an additional 55infill samples were generatedto improvethequalityofpredictions.Theinfillsampleswereselected basedontheuncertaintyestimatorforaKrigingmodeltrainedto predict Mre f fromthe initialsample nacelledesigns.For5ofthe

Fig. 7.Overview of the RSM construction process.

adverse aerodynamics.Thetotal setofCFDresultscontained285 designs. Foreach ofthesedesigns, the methoddescribed in

Sec-tion 2.1.1 was used to determine drag at the combinations of M

and MFCR shown in Fig. 5. This data was then used to train a drag map interpolation model (Section 2.1.2) for each drag map to compute the scaling references (cD,re f, Mre f) and normalized

drag maps (Section 2.1.3). For 6 of the 285 sample designs, the drag map interpolationmodel generationfaileddueto highly er-ratic drag distributions associated withseverely adverse designs, andtheywere removed fromtheset.The remainingset contains 279designswith196CFDsolutionsperdesign,comprising54684 CFDsolutions.

Foreach design,thedragmap processing(Section2.1.3)yields thescalingreferences(Mre f andcD,re f),aGaussianprocess to

in-terpolatethedragmap,andthedragmapitself,normalizedtoM_T,

MFCRT andc_D.

2.2.2. ConstructionofRSM

AschematicoverviewoftheprocessbywhichtheRSMis con-structedisshowninFig.7.Itisexplainedindetailbelow.

Afterestablishingthescalingreferencesandcomputingthe nor-malizeddragmaps foreach design inthe database, two Gaussian processmodelsaregeneratedtointerpolatethescalingreferences (Mre f andcD,re f)acrossthedesignspace.Inbothcases,thechoice

ofcorrelationandregressionmodel,aswellastheconstantnugget valuewereoptimizedtominimizetheRMSofthepredictionerrors obtained through leave-one-out (LOO) auto-crossvalidation. LOO auto-crossvalidationisthebest-practicemethodtoobtainan unbi-asedmeasureoftheaccuracyofmachinelearningmodels[27].

The ability topredict the scaling references (Mre f andcD,re f)

at any point in the geometrical design space makes it possible toconvertbetweendimensional (M,MFCR) andnormalized (M_T,

MFCRT)operating conditionsforanydesign.Thisallows theRSM

generationprocess to workon thenormalized dragmaps,whose drag-rise slopes and subcritical drag levels are aligned, as illus-tratedinFig.6c.Thepartofthedataforwhichthedragprediction mustbeaccurateisthesubsetwherecD

≈

cD,re f,andthuscD

≈

1.

Thisisonlyarelativelysmallportionofthefullrangeofc_D inthe inputdata.

FittinganRSMtodirectlyinterpolatethenormalizeddragmap dataindesignspaceissimilar togeneratinga drag map

interpo-lation model (Section 2.1.2), as there are regions of strong non-linearities andregions in which thedata changes little butmust bemodelledveryaccurately.Tosolvethisproblem,thenormalized drag map data was decomposedfurther, intoa meannormalized drag map, c_D

=

f

M_T

,

MFCRT

,and a deviationfrom thismean. Thiswas doneby usingtheGaussianprocess modelsforeach in-put drag map to interpolateall normalized drag maps onto one regular grid in M_T and MFCRT, and compute the mean normal-izeddragoneachofthegridpoints.Topreventdatafromdesigns withparticularlyadverseaerodynamicpropertiesfrominfluencing theresult,andtoimprovequalityofthemethodfornacelleswith beneficialaerodynamics,inputdatapointswereignoredbythe av-eraging process if drag was more than two standard deviations abovethemeannormalizeddrag(c_D)atthesamenormalized op-eratingcondition (M_T

,

MFCRT).Theaverageddragdatawere then used to generate a drag map interpolation model in normalized coordinates (modelling c_D

=

f

(M

_T

,

MFCRT)). Similar to a custom regression model for a Gaussian process, this allows to split out largesystematictrendsfromthedata.

Aftercomputingthemeannormalizeddragmap,thedrag quo-tientQcD (Eq. (13))wascomputedforeachnormalizedinputdrag

map.QcD isafunctionofM_T andMFCRT andrepresentsthe

quo-tientbetweenthedragofeachsamplednacelledesigntothedrag that wouldbe obtainedby re-dimensionalizingthemean normal-izeddragmap.

QcD

M_T

,

MFCRT

=

cD

M_T

,

MFCRT

−

1

+

cD,re f c_D

M_T

,

MFCRT

−

1

+

cD,re f (13)

Fig.8illustratestheprocessusingtheexampledrag-risecurves shownin Fig.6.The fourexampledrag-risecurves inFig.8a are decomposed intothe mean drag-risecurve of all nacelledesigns in thedataset (Fig. 8b),andthe drag quotientQcD (Fig. 8c). The

meandrag-risecurvecontainsmuchofthesystematictrends.With these systematiccomponents removed, the individual differences between thedrag-rise curvesmake up a much larger proportion of the overall signal in the QcD data (Fig. 8c), compared to the

normalizedcurves(Fig.8a).

The last componentof theoverall RSM method isa Gaussian process model which interpolates QcD, asa function of all eight independent variables: six design variables (g1 through g6) and thenormalizedoperatingconditionsM_T andMFCRT.Itwas found that thisGaussian processmodelproduced themostaccurate re-sultswhenchoosingacubiccorrelationmodel[28].Noregression function was used, as systematic trends have already been ex-tracted from the data as far as possible. Each input data point was assigned thesame nuggetasfor thedrag map interpolation model described in Section 2.1.2. Since there are as many input pointsasthereareCFDsolutionsintheinputdataset(ca.55,000), thisrequiresasignificantamountofcomputationalworking mem-ory.A setofthree filterswasapplied beforefittingthemodel,to remove data from the set which may make prediction of desir-able performance less accurate, and at the same time reducing the memory requirements per nacelle design to allow inclusion of a larger number of sample designs. The first filter removes all data above Mre f (Eq. (14)). Since M_T

=

1 maps to Mre f and

Mre f

>

MD R,thisonlyremovesdataabovetheMachnumberrange inwhichaccuratedragpredictionisrequired.Thesecondfilter re-movesalldatapointsatwhichQcD islargerthanthree(Eq. (15)), i.e. drag data points which is more than three times ashigh as would be expectedbased on the meannormalized drag and the referencedrag atthesame normalizedoperatingconditions(M_T,

MFCRT). Thejustificationforthisfilteristhatdesigns which

nor-Fig. 8.(a) Normalized example drag-rise curves, (b) mean drag-rise curve, (c) theQcDfunction for the example drag-rise curves.

malized conditions are not suitable to operate at this condition, thereforeanaccuratepredictionisnotrequired.Itisalsoassumed thatthereremainsenoughdataintheinputsetwithQcD

<

3

.

0 at

neighbouringoperating conditionsto informthemodel that QcD

is very large. The third filter removes all data points from the input set witha nugget value above 2.0 (Eq. (16)). According to Eq. (8), this meansthat theexpected relative uncertaintyon the input value is greater than

√

2 (i.e. larger than 140% of the in-putvalueitself),andtheinformationlostfromremovingtheinput wouldnothavenotableimpactonthemodelprediction.

M_T

<

1

.

0 (14)

QcD

<

3

.

0 (15) n

<

2

.

0

⇒

σ

QcD

QcD

<

√

2 (16)

2.2.3. Draginterpolationmethod

Themodelbuildingprocedure generatesone Gaussian process modeltopredict each ofthescaling references(Mre f andcD,re f)

across the design space, one Gaussian process model to predict thenormalizedmeandragasafunctionofthenormalized operat-ingcondition,specifiedby M_T andMFCRT,andaGaussianprocess

modelto predict QcD (Eq.(13)), asa functionof thenormalized operatingcondition andthe sixdesign variables. To predict drag fora givennacellegeometryatagivenoperatingcondition, aset ofsixdesignvariablesandanoperatingcondition, determinedby

MandMFCR,mustbeprovided.First,thescalingreferences(Mre f

andcD,re f,Eqs. (10), (9))arepredictedfromthedesignvariables,

usingthe Gaussianprocessmodels forcD,re f and Mre f.The

oper-atingcondition(MandMFCR)isconvertedtonormalizedvariables (M_T and MFCRT), using Eqs. (12) and (5), and the interpolated

Mre f. With the normalized operating condition, the normalized meandrag mapmodel (Fig.8b) canthen be evaluatedto predict themeannormalizeddrag coefficient, c_D.The drag quotientQcD

ispredictedfromthe samevariables(Eq. (13),Fig. 8c).From this data,the predicteddrag coefficientcan becomputedbyinverting Eq.(13) andcombiningwithEq.(11) intoEq.(17).

cD

=

c_D

−

1

+

cD,re f

QcD (17)

TopredictMD R foragivendesign,atagivenMFCR,onecould po-tentiallygenerateaseparateRSMtomodelMD R

=

f

(MFCR,

g1

...g

6

)

directly. This was not pursued because the results of such an independent RSM would not necessarily be consistent with the derivativesofthepredicteddrag,andbecause MD R isnot guaran-teedto bea continuous function oftheother seven independent variables,andthere maybe morethanone uniqueMachnumber foragivendesignatoneMFCRwhichmeetsthedrag-risecriterion (Eq.(1)).Instead, MD R isfound by predictingthe drag-risecurve foranacelledesignanditerativelyevaluatingdraggradienttofind theMachnumberwhichsatisfiesEq.(1).

3. Results

Several tests of components of the RSM and of the overall model were carried out. Aninvestigation into the distribution of nacelledragpredictionerrorsacrossdifferentoperatingconditions was also conducted. Predictionsfromthe drag map interpolation model were testedagainst a high-resolution drag map generated from CFD. The interpolation models for the scaling references (Mre f and cD,re f) were tested through LOO auto-crossvalidation,

and the overall RSM was tested against an independent CFD dataset,whichallowsforstatisticalcomputation ofthemodel ac-curacy. Finally, the distribution of nacelle drag prediction errors fordifferentoperatingconditionswascharacterizedbycomparing predictionsfromtheRSMandCFDforseveralexamplenacelle de-signs.

3.1. Dragmapinterpolation

Both the computation of the scaling references (cD,re f and Mre f) fromtraining data andthe evaluationofmean normalized

drag,c_D,inthedrag predictionprocess(Section2.2.3)depend on thedragmapinterpolationmethoddescribed inSection 2.1.2.The drag map interpolation thus needs to providenot justaccurately interpolated drag butmustbe consistent enough to provide reli-ableestimatesofthederivativeofdraginregardto Machnumber. To test the drag map interpolation model, a CFDdrag map with a resolution of 125 Mach numbers and30 values for MFCR was generatedforone specific nacelledesign. Thiswas inadditionto a drag map withstandardresolution of28Mach numbersand7

MFCRs,generatedforthesamenacelledesign.

Fig. 9 shows contour levels of drag obtained from CFD data sampled at high resolution (Fig. 9a) and standard resolution (Fig.9b).Thecomparisonofdrag-risecriteria(solidlinesinFig.9b) basedoneachdatasetshowsanoffsetof0.0024forallMFCR

>

0

.

6, whenusingfirst-orderfinitedifferencestocompute thedrag gra-dient (Eq. (1)). The same offset is observed with respect to the criterion for Mre f (Eq.(10),dashed linesin Fig.9b).Another vis-ible differencebetweenbothmapsaretheoscillations oftheline for dcD

/dM

=

0

.

1 in the high-resolution data (Fig. 9a). A direct comparison between the two maps found that these originate with small-scale oscillations in the CFD results for nacelle drag atMFCR

<

0

.

55, M

>

0

.

6,with anamplitude of lessthan0.5% of nacelle drag. At these low MFCR values, drag in transonic flow is very sensitiveto smallchanges in thehighly loadedboundary layerinteractingwithasonicshock.Thiseffectcausesareduction innumericalconvergenceandamplifiesitseffectonthecomputed nacelledrag. Althoughthe amplitudeoftheseoscillations isvery small, thecalculationofdcD

/dM

isaffectedduetothehigh sam-plingrateusedforthehigh-resolution dragmap, whiletheeffect is notnoticeable atthe normalsamplingresolution.Both the in-accurate MD R andMre f obtainedbylinearinterpolationfromthe

Fig. 9.Dragmapfrom(a)high-resolutionCFDdata(125×30 samples)and(b)standardresolution(28×7).Dragmapsinterpolatedfromstandardresolutiontohighresolution using(c)ordinaryKriging(OK)and(d)theproposednewdragmapinterpolationmethod(Section2.1.2).Differencebetweenhigh-resolutiondataandmapsinterpolatedby (e)OKand(f)theproposednewmethod.SolidlinesmarkdcD/dM=0.1,dashedlinesmarkdcD/dM=0.5.Blacklinesrepresentdatafromthehigh-resolutionCFDdrag map.Symbolsdenotesamplinglocations.

fromthehigh-resolutiondrag mapconfirm therequirementfora moresophisticatedinterpolationmodeltoobtainaccurate,smooth draggradients.

Totestthedragmapinterpolationmethodusedbythenacelle dragpredictionRSM,theCFDnacelledragdataatstandard resolu-tioninFig.9bwasinterpolatedtotheoperatingconditionswhich hadbeen sampledin the high-resolution drag map (Fig. 9a). For comparison,thiswasdonewithanordinaryKriging(OK)process, inadditiontothemethoddescribedinSection2.1.2.

Fig. 9a shows the results of the naïve application of OK to modelcD asafunctionofMandMFCR,basedonthestandard res-olutiondragmapCFDdata.Thisapproachdiffersfromthemethod described in Section 2.1.2 in that it does not use a regression model(“constantregression”),andusesaconstantnuggetvalueof 4

×

10−6_{, corresponding tothe baseline nugget employed bythe}

drag map interpolation method. The resulting interpolated drag map shows that the ordinary Kriging (OK) model produces con-siderable errors even at low Mach numbers, up to the drag-rise Mach number (Fig. 9c). At the sametime, the high-dragportion of the drag map is comparably well reproduced, including the curve fordcD/dM

=

0

.

5. The CFD-derived andOK-derived curves fordcD

/dM

=

0

.

5 showaMach numberdifferenceof 0.01atthe

maximum MFCR, whichindicates a lessaccurate interpolation of draggradientthanachievedbylinearinterpolation(Fig.9b). Com-pared to this, the lines indicating dcD

/dM

=

0

.

1 (Eq. (1)) bear

almost no resemblance to the original data, as the oscillations in the low-drag range of the drag map cause the gradient of

dcD

/dM

=

0

.

1 to be exceeded frequently, even at M

<

0

.

3. Rela-tive interpolation errors (Fig. 9e) reach orexceed 10% inseveral places, even at conditions where there is little variation in the drag data, but the amplitude of errors does not increase in the high-drag regions of the drag map. The large errors at low-drag operating conditions are the consequence of the model adapting to the highdrag gradients andstrong non-linearities inthe drag data at Mach numbers above MD R. The ordinary Kriging inter-polation does not allow useful predictions of drag or drag gra-dients at the relevant operating conditions. The fact that higher gradients are represented more accurately by OK confirms that the OK model adjusts to the stronger gradients and larger drag amplitudes found in the data above drag-rise Mach number, to the detriment of accuracy at conditions with small drag gradi-ents.

The dragmap interpolation method described in Section 2.1.2

produces a much more accurate approximation of the high-resolution drag map, based on the standard resolution data (Fig. 9d). The lines for dcD

/dM

=

0

.

1 and for dcD

/dM

=

0

.

5 are in excellent agreement. The error is small enough that drag-rise Mach numbers (MD R, Eq. (1)) can be determined from the

in-terpolated datawith an erroroflessthan 10−3_{, atMFCR}

>

₀

.

_55,

Fig. 10.Validationagainstanindependentdataset,predictionsfromRSMcomparedtoCFDresults.(a)Drag-riseMachnumberatMFCR=0.75 fromasimpleGaussian processmodel;(b)dragatcruiseconditions(Eq. (2) fromasimpleGaussianprocessmodel;(c)and(d)usethecombinedRSMdevelopedinthispaper.

margins are significantly lower than with either linear interpo-lation or OK, and also an order of magnitude below the errors requiredforthenacelle drag predictionRSM.The differences be-tweentheinterpolatedmap(Fig.9d)andthehigh-resolutiondrag map from CFD (Fig. 9a) are below 1% at all conditions where

M

<

Mre f (Fig. 9f). The error only exceeds 1% at the highest sampled Mach numbers which have no relevance for the appli-cationathand.Thisistheintendedresultofde-prioritizinginput datapoints withvery large drag gradients. The small-scale oscil-lations visible around M

=

0

.

7 at the lowest MFCRs are not due totheinterpolationbutoriginatewiththehigh-resolutionCFD,as alreadyexplainedinthediscussion toFig. 9a andb.Ina prelim-inarydesign context, it is desirable that the interpolation model produces smooth gradients in place of any small-scale oscilla-tions.

3.2.Interpolationofscalingreferences

Theinterpolation models forthe scaling references (Mre f and

cD,re f, Section 2.1.3), were tested by leave-one-out (LOO)

auto-cross-validation[27]. The RMS ofthe predictionerror was found tobe2

.

6

×

10−4 _forc

D,re f.Withtypicalvaluesof0.025ofcD,re f,

thiserroriswellbelow1%ofthecD,re f value.Thismeansthatthe

interpolationmodelforcD,re f doesnotaddanysignificantamount

ofuncertaintytotheoverallmethod.

The LOO validation for the Mre f model found an RMS of 4

.

7

×

10−3_{.Fora representative caseof M}

re f

=

0

.

9, MD R

=

0

.

85,

thisleadstoanerrorofabout5

.

5

×

10−3 _{intheestimateofM} D R.

Thisisanacceptable result, andit issignificantlybetter thanthe uncertaintyofnacelledrag-risepredictioninpublishedpreliminary methods[29],whichisaround0.02.

3.3. Fullmodelvalidation

3.3.1. Validationdatasetandvalidationcriteria

Generating a Gaussian process model for QcD requires a

no-table amount of computational effort, therefore it is impractical toconductLOOcross-validationfortheQcD modelorforthefull RSM.To obtain reliable indicatorsof theaccuracy ofthe full na-celle drag prediction model (Section 2.2.3), an independent CFD dataset was used. This consisted of 93 designs within the same designspace(Table1),sampledaccordingtoaLatinhypercube dis-tribution.ThesenacelledesignswereanalysedusingthesameCFD procedureastheinputdatafortheRSM.

Thevalidationregardedthepredictionoftwoperformance met-ricswhicharedefinedinSection1.2.Thefirstmetricwasdrag-rise Mach number(MD R,Eq. (1)) atMFCR

=

0

.

75. The second metric was dragattheassumedrepresentativecruisecondition (Eq. (2)), which is specific to each nacelle design.Each of thetwo perfor-mancemetrics wasextractedfromtheCFDvalidationdataset, us-ingthedrag mapinterpolation methoddescribed inSection2.1.2. ThesamemetricswerethenpredictedfromthefullRSM,and com-paredtotheresultsfromtheCFDdataset.

3.3.2. Predictionofperformancemetrics

To compare theresults of the RSM model to the output ofa more conventional process, the Gaussian process used to predict

QcD wastrainedondataofdragcoefficient(cD)directlyasa

func-tionofM,MFCR,andthesixdesignvariables.Sincethefulldataset istoolargetotrainaGaussianprocesswiththeavailableworking memory, thismodel usedthe reduceddataset left after applying thefiltersinEqs.(14)–(15).Thisintroducessomeinformationfrom thecompleteRSMtothesimplecomparisonmodel,andgivesitan advantage overan entirelynaïvemethodwithfewer designs and

Fig. 11.ComparisonofCFDresultsandpredictionsfromRSM.LinesshowMD RasafunctionofMFCRcomputedfromCFDdata(solid),andpredictedfromtheRSM(dashed). (a),(b)DragmapsfromCFD;(c),(d)dragmapsforthesamenacelles,fromRSM;(e),(f)erroroftheRSMpredictionrelativetotheCFDresults.PositivecDindicatedrag over-prediction.

nofiltering ofundesirableinput data.The nuggetforthis simpli-fied RSM was set to a constant value of 4

×

10−6_{, the baseline}

nuggetusedfortheQcD andthedragmapinterpolationmodels. Validation was conducted against the independent validation dataset of 93 nacelle designs (Section 3.3.1). Both the simplified and the full RSM were used to predict the performance metrics

MD R and cD,cruise (Eqs. (1), (2)).Fig. 10showsthe resultsofthe

validation for both models. To quantify the goodness of fit, the standarderrorwascomputedforeachpredictedvariableforeach model,accordingtoEq. (18).

σ

X

=

1 n n

i=1

Xpredicted

−

XC F D

2 (18)

For the simplified RSM, the standard error for MD R is 0.023, with a considerable number of outliers in the validation graph (Fig.10a).Withinthecontextofthisstudy,thisisanunacceptably largeerror,asthemajorityofmodelevaluationswouldresultin er-rorslargerthanthetypicaldesignmarginof0.02betweenMcruise

and MD R. The standard error forcD,cruise (Eq. (2))is 2

.

3

×

10−3

or9

.

5% of thecorrect cruisedrag (see also Fig.10b). In contrast to this, thefull response surface method presentedinthis paper achieves a standard error for drag-riseprediction of 0.014, with feweroutliers(Fig.10c)thanobservedintheresultsfromthe sim-plemodel(Fig.10a) andastandarderrorforcruisedragprediction of3.6%,almost athirdofthe errorachievedbythesimple Gaus-sianprocess.Bothresultsclearlysurpasstheerrormarginsquoted

forestimationofcD andMD R withthepreliminarynacelledesign

methodpublishedbyESDU[29],of8%onsubcriticalnacelledrag, 20%onwavedragatcruiseand0.02onpredictionofMD R.

85%ofallobservederrorson MD R aresmallerthan0.014,and in 75% of all cases the error is smaller than 0.01. With a nor-mally distributed error, one would expect to haveonly 68.1% of all sampleswithinthestandard deviation.Thissuggeststhat out-lierscontributesignificantlytothestandard error,andthatinthe majority ofcases,predictionsare alreadymore accuratethan the standard errorindicates. Theperformanceoftheresponsesurface modelforpredictingcruisedragofnacelles(Fig.10d)ismorethan sufficient fora preliminary design application, where average er-rorsoftheorderof10%areacceptable.Thislimitisonlyexceeded foroneofthe93validationdesigns.

3.3.3. Predictionofdragmaps

To assess the ability to predict the properties of likely na-celle designs, as opposed to the random designs in the valida-tiondataset,andtotestthepredictionaccuracyofdragprediction acrossdifferentoperatingconditions,twoexamplenacelle geome-tries were generated. These nacelles were chosen fromthe valid design space to have desirable performance characteristics. This wasdonetoevaluatethemodelperformanceforcaseswhichcould reasonably be theresults of a designprocess, andfor which de-taileddrag predictionisrelevant,ratherthanforarbitrarydesigns whichare oflessrelevanceandalsomorelikelyto have hard-to-interpolateextremedragcharacteristics,suchasshowninFig.2c. Fig.11showsthedifferencebetweentheRSMpredictionandCFD

resultsforthetwonacelledesigns.AcomparisonofFig.11a,bto Fig.11c,dshowsthat thegeneralshapeofeach dragmapis cor-rectlypredictedbythe RSM,includingthecharacteristicsof MD R

asafunctionofMFCR.Nooscillationsorsystematicerrorsare no-ticeableinthe predicteddrag maps.Theerrormapsin Figs.11e, f show that the accuracy of drag prediction at subcritical Mach numbersisgenerallywithin 1%. AtMach numbersclose to MD R, thisdifference increasesbutgenerallystays below10% atall op-eratingconditionsbelowMD R.ThefactthataboveMD R,theerror increasesandcanreach20%,isacceptable forpreliminarydesign, andisinpartalsointendedconsequenceofprioritizingaccuracyat cruisecondition overhigh-drag conditions.Thefact thatthe pre-dictionerrorsshowninFig.11e,fchangeverygraduallywith op-eratingconditionsbelow MD R means thatthe RSMissufficiently consistenttocompute drag gradients.The gradientsofdrag error increaseaboveMD R,butarestillsmallenougharounddrag-riseto

predict MD R withgood accuracy,withan error oflessthan0.01 forbothnacelledesigns,forMFCR

>

0

.

55.

4. Conclusions

The problem of nacelle drag prediction from RSM presents a unique combination of challenges which prevent the application of standardized methods. One challenge is that the data cannot be sampled isotropically, because not just drag but gradient of dragoverMachnumberneedstoberepresentedaccuratelyinthe sampledinputdata.ThispreventstheuseofaglobalLatin hyper-cubesampling(LHS).Anotherchallengeisthatboththeinputdata andtheaccuracyrequirements areinhomogeneous. Atconditions oflow drag andlow drag gradients,modellingmust be accurate, whilethepresenceofhighdragsandlargegradientsatother con-ditionsneedstobeindicatedbutnotmodelledaccurately.Evenfor asinglenacelledesign,anordinaryKrigingapproachfailstomodel theinhomogeneouscharacteristicsofthedata.Thethirdchallenge is that the model is effectively used as a multivariate predictor sincetheimmediateoutput,nacelledrag, isusedtocompute the gradientofdragover Machnumber,whichisthen usedto deter-minethedrag-riseMachnumber.

The presented method decomposes the problem into several components which are based on Gaussian processes. It exploits self-similaritiesbetweendifferentdragmapsandorthogonal prop-erties of designs, such as drag-rise Mach number and the drag levelatlow Mach numbers. Theseare then interpolated,per de-sign,byseparatemodels.Anotherstrategythatwasappliedisthe consequentuseoffilteringandnuggetstofocusaccuracyontothe mostrelevantparts ofthe data.This allowsprediction ofnacelle dragwithastandarderrorof3.6%atcruisecondition,whilevery sharpdrag changes outside ofthe plausibleoperating rangeof a nacelledesignareonlyapproximatelymodelled.Atthesametime, themodelretainstheimportantabilitytoindicatetherangeof vi-ableoperatingconditions.Predictionsareconsistentenoughto ob-tainsmooth firstderivativesofnacelledrag withrespect toMach number.Thisallows predictionof drag-riseMach numberwitha standarderrorof0.014.Thenewmethodisanotableimprovement on previously published methods for preliminary nacelle design, whichclaimstandarderrorsof0.02ondrag-risepredictionand8% onthepredictionofnacelledragatcruisecondition.

Conflictofintereststatement

Theauthorsdeclarethatthereisnoconflictofinterest.

Acknowledgements

TheauthorswouldliketothankChristopherSheafandNicolas GrechfromRolls-RoycePLCforprovidingadvice.

Thisworkwasco-fundedbyInnovateUK.

Datastatement

Due to confidentiality agreements with commercial research collaborators,supportingdataforthisstudycannotbeshared.

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