A look-up table method based on unstructured grids and its application to non-ideal compressible fluid dynamic simulations

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Delft University of Technology

A look-up table method based on unstructured grids and its application to non-ideal

compressible fluid dynamic simulations

Rubino, A.; Pini, M.; Kosec, M.; Vitale, S.; Colonna, P.

DOI

10.1016/j.jocs.2018.08.001

Publication date

2018

Document Version

Final published version

Published in

Journal of Computational Science

Citation (APA)

Rubino, A., Pini, M., Kosec, M., Vitale, S., & Colonna, P. (2018). A look-up table method based on

unstructured grids and its application to non-ideal compressible fluid dynamic simulations. Journal of

Computational Science, 28, 70-77. https://doi.org/10.1016/j.jocs.2018.08.001

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application

to

non-ideal

compressible

ﬂuid

dynamic

simulations

A. Rubino

a

_,

_M.

_Pini

a,∗

_,

_M.

_Kosec

b

_,

_S.

_Vitale

a

_,

_P.

_Colonna

a

a_Delft_University_of_Technology,_Aerospace_Engineering_Faculty,_Propulsion_and_Power,_Kluyverweg_1,₂₆₂₉_HS_Delft,_The_Netherlands b_Stanford_University,_Aeronautics_and_{Astronautics,}_Stanford,₉₄₃₀₅_CA,_United_States

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received28May2018

Receivedinrevisedform18July2018 Accepted1August2018

Availableonline9August2018 Keywords: Look-uptable Trapezoidalmap Unstructuredmesh Thermodynamicmodeling Interpolation Real-gasﬂows

Non-idealcompressibleﬂow NICFD

CFD

a

b

s

t

r

a

c

t

Fastandaccuratecomputationofthermo-physicalpropertiesisessentialincomputationallyexpensive simulationsinvolvingfluidflowsthatsignificantlydepartfromtheidealgasoridealliquidbehavior.A look-uptablealgorithmbasedonunstructuredgridsisproposedandappliedtonon-idealcompressible fluiddynamicssimulations.Thealgorithmgrantsthepossibilityofafullyautomatedgenerationofthe tabulatedthermodynamicregionforanyboundaryandtousemeshrefinement.Resultsshowthatthe proposedalgorithmleadstoacomputationalcostreductionuptooneorderofmagnitude,whileretaining thesameaccuracylevelcomparedtosimulationsbasedonmorecomplexequationofstate.Furthermore, acomparisonoftheLuTalgorithmwithauniformlyspacedquadrilateraltabulationmethodresultedin similarperformanceandaccuracy.

1. Introduction

Theaccurateestimationofthermo-physicalpropertiesof flu-idsisessentialfor manyengineeringandscientificapplications, anditrequirescomplexmodelsincasethebehaviorofthefluid departsfromthatoftheidealgasoridealliquid.Fluids exhibit-ingnon-idealbehaviorareinvolvedinvarioustechnologiessuchas advancedpowerandpropulsionsystems,refrigerationandair con-ditioningsystems,oilandgasprocesses,etc.[1–4].Inthesecases, theevaluationofthermo-physicalpropertiesisoftennecessaryfor systemdesignandperformanceevaluationortosimulatetheflow behaviorwithincomponents.Fluiddynamicsimulationsofvapors innon-idealstatesarealsoemployedinmorefundamentalresearch (see,e.g.,Ref.[5])andthebranchoffluidmechanicsdealingwith thistypeoffluidflowswasrecentlytermednon-idealcompressible fluiddynamics(NICFD)[6].

Thecomputationalcostassociatedwithﬂuidthermodynamic modelsexpressedintermsofequationsofstate(EoS)canbecome alimitingfactorifaccurateestimationsareneededin combina-tionwithexpensivesimulations.Thisisthecase,e.g.,inthedesign andoptimizationofindustrialcomponents[7]orincomputational

∗ Correspondingauthor.

E-mailaddress:M.Pini@tudelft.nl(M.Pini).

ﬂuiddynamics(CFD)[8].Inordertodecreasethecomputational timerelatedtothecalculationofthermo-physicalﬂuidproperties, whilemaintainingasatisfactorylevelofaccuracy,look-uptable (LuT)methodsareconvenientandwidelyadopted[9].

ALuTmethodconsistsofthreebasicparts:1.Thetabulation ofadiscretesetofvaluesofthermodynamicproperties pertain-ingtostatesgeneratedwithagivenEoS-basemodel;2.Asearch algorithm;3.Aninterpolationmethod.Sincethetabulationis per-formedonlyonce,atpreprocessinglevel,this wayofevaluating fluidthermophysicalpropertiescangreatlyreducethe computa-tionaleffortifthemodelsarebasedoncomplexEoS,providedthat theassociatedsearchalgorithmisefficient.Furthermore,the inter-polationtechniquemustbecarefullychoseninordertoachievea satisfactorylevelofaccuracy,whichisafundamentalrequirement toguaranteeconvergenceinCFDsimulationsandaccuracyofthe finalresult.LuTmethodsbasedonastructuredmeshof thermody-namicregionofinteresthavebeendocumentedintheliterature

[10,11,9,12]. In order to increase the accuracy and decreasing thenumberofdiscretizationpointsforthermodynamicregionof interest,algorithmsbasedonadaptiveCartesianmeshhavebeen proposed[13,14].However,theuseofquadrilateralgrids,in com-binationwithlocalreﬁnement,canleadtolocaldiscontinuities andpoorinterpolationaccuracyofpropertiesintheproximityof smoothboundaries[15,13].Thiscanespeciallyoccurfor

proper-https://doi.org/10.1016/j.jocs.2018.08.001

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Fig.1.ThermodynamictriangularmeshforthesiloxaneMDM.(a)Regularthermodynamicmesh.(b)Reﬁnedthermodynamicmesh.

tiesreconstructionorforinterpolationclosetothevapor–liquid saturationline.

Thestudydescribedhereresultedinug-LUT,anewLuTmethod basedonmeshingthermodynamicregionofinterestbymeansof unstructuredtriangulargrids.Thegenerationofmultidimensional thermodynamictablesisfullyautomated,evenincasemultiple fluidphasesneedtobecomputed.Unstructuredgridsallowfor meshrefinements,a valuablefeatureincaseofstrongproperty variations,whichoccurinproximityofthevapor–liquidcritical point,and ofthesaturationlineingeneral.Theug-LUTmethod isapplicablealsotomulti-componentfluids.Asearchalgorithm basedonatrapezoidalmapofthetabulatedregioncontributes sig-nificantlytoitscomputationalefficiency.Theug-LUTmethodfor thermo-physicalpropertycalculationsisimplementedwithinthe open-sourcecodeSU2[16–18]anditsverificationispresentedby meansofparadigmaticCFDtestcasesofincreasingfidelity.Finally, theug-LUTalgorithmiscomparedtoastructured-basedLuTwith theaimofassessingitscomputationalcostandmemory require-ments.

2. Method

2.1. Generationofthermodynamicmesh

Anunstructuredmeshisgeneratedforthermodynamicdomain of interest. A 2D gridgenerator [19], based onthe Advancing-Delaunayfront method,isusedin this study.Theadoptedgrid generatorallowsforlocalreﬁnementsinselectedregionsof ther-modynamic domain.Fig. 1 shows examplesof thermodynamic meshes for siloxane MDM (octamethyltrisiloxane,C8H24O2Si3),

generatedbyselectingTandlog()asinputstatevariables.The useofunstructuredmeshincombinationwithlocalrefinementis proposedhereasaneffectivealternativetostructuredand quad-treegrids.Quad-treebasedalgorithmsareefficientforcontrolling themeshrefinementlevel,butcansufferfromthefollowingissues

[15,13]:(i)storingandretrievingthemeshconnectivityassociated totherecursivetreestructure;(ii)hangingnodesatdifferentsizes cellsinterfaces,ifcontinuouspropertyreconstructionisrequired; (iii)additionalinterpolation,triangulationoracurvilinearmesh systemmightbenecessarytoreconstructsmoothboundaries(e.g. vapor–liquidsaturationline).

Oncethemeshpatches aregenerated usinganysuitableset ofstatevariables(twoiftheﬂuidispure),allotherneededﬂuid thermo-physicalpropertiesarecomputedateachmeshnodewith anappropriatethermodynamiclibrary.Thecurrent implementa-tionoftheug-LUTmethodmakesuseofanexternalthermodynamic library[20],whichembedsalargevarietyofmodelsbasedon com-plexequationsof state(EoS).Asanexample,Fig.2 reportsthe

Fig.2. TabulatedpressurecontoursobtainedforthesiloxaneMDM.

pressurecontourforsiloxaneMDM,obtainedusingamodelbased onanEoSintheSpan-Wagnerfunctionalform[21].

2.2. Searchalgorithm

Oncethesetofthermo-physicalpropertiesfortheselectedﬂuid arestoredintabularform,asearchalgorithmisusedtoretrieve thebestapproximationofthequerystateorpoint.Apoint loca-tionalgorithmbasedonatrapezoidalmaphasbeenadopted[22]

becauseofthefollowingconsiderations:(i)thesamegeometrical connectivityisusedforallthesearchpairs.Thisisespecially ben-eﬁcialbecausetheconnectivityhasnottoberecomputedforeach searchpairanditcanbeusedtosearchinhighlyskewed thermody-namicplanes.Asanexampleforthis,iftheinitialthermodynamic meshisbuiltforthe(P,)plane,theresultingmeshonthe(h,s) planewillbehighlyskewed;(ii)searchingforthetriangle contain-ingthequeryvectorbyresortingtoalgorithmsthatdonotusethe meshconnectivityinformation(e.g.kd-tree)onanirregularand highlyskewedgridcanleadtoinaccurateinterpolation;(iii) trape-zoidalmapsworkforgeneralpolyhedra.Thesearchalgorithmcan beusedtoswitchbetweendifferentmeshzones,characterizedby apolyhedronoutline.Thisfeatureallowstoavoidamappingfor gridsnotconformingwitharectangularthermodynamicdomain

[13].

GiventhesetSofntriangularmeshedges,thetrapezoidalmap T (S)isbuiltaccordingtothefollowingsteps(seeFig.3):

1.auniquesetofedgesandthecorrespondinglistofitsx coordi-natesiscreatedbyﬁlteringoutduplicates;

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Fig.3.Schematicrepresentationofthetrapezoidalmapforanunstructuredgrid.

2.theintersectingedgesareassociatedtoeachband; 3.theedgesineachbandaresorted.

Thetrapezoidalmap T (S)iscreatedina preprocessingstage foreachthermodynamicsearchpair,e.g.,(h,s),(,

v

),etc.Amore detaileddescriptionofthetrapezoidalmapalgorithmcanbefound inAppendixA.

Themeshsimplex,containingthequerypointqwithinT (S),is identiﬁedaccordingtothefollowingprocedure:

1.thexcoordinateofthequerypointisobtainedwithabinary searchforitscontainingband;

2.withinthecontainingband,theedgeaboveandtheonebelow thequerypointareidentiﬁed;

3.thesimplexcontainingthequerypointissingledoutbyusing theedge-to-faceconnectivityofthetwoedgesselectedduring thepreviousstep.

ThequeryalgorithmisalsodetailedinAppendixA.

2.3. Interpolationmethod

Atwo-dimensionallinearinterpolationproblemcanbewritten as

f(x,y)=N_i=1wigi(x,y)=WTG(x,y)=GT(x,y)W, (1)

whereG(x,y)isaninterpolationbasiswhichtransformsthequery coordinatesx,y(rawfeatures)intoNlinearlyindependent inter-polationfeatures.AnexampleofapolynomialbasisfunctionG(x, y)onatriangle(threepoints,N=3)isgivenby

G(x,y)=[1,x,y]T. (2)

In the implemented thermodynamic look-up-table, the two-dimensionalcoordinatesarethermodynamicpairssuchas(,u), (P,T),(h,s).ThischoiceofG(x,y)wasfoundtobesufﬁciently accu-rate,providedthatarelativelyﬁnemeshforthelook-up-tableis used.ThevectorofweightsWisfoundthrough

A=

⎡

⎢

⎣

GT_(x 1,y1) . . . GT_(x i,yi) . . . GT_(x N,yN)

⎤

⎥

⎦

, F=

⎡

⎢

⎣

f(x1,y1) . . . f(xi,yi) . . . f(xN,yN)

⎤

⎥

⎦

, AW=F (3) canbewrittenas G(xq,yq)TW=FTV, (4)

wheretheinterpolationweightsVarenowtheadjointsolutionof

ATV=G(xq,yq). (5)

Theequivalenceofthetwoformulationscanbeputintoevidence by[23]

VTF=VT(AW )=(ATV )TW=G(xq,yq)TW. (6)

The weights V can be computed once for a given query point andreusedtocalculatethedifferentthermodynamicproperties ofinterest.Forexample,iftwelvepropertiesneedtobe interpo-lated,only onematrix-multiplicationis requiredpereachmesh triangle,insteadofthetwelvewhichwouldbeneededwiththe primalinterpolationmethod.Additionally,sincethe(AT₎−1_matrix

dependsonlyonaprioriestablishedvalues,itcanbecompletely pre-computedwithoutadditional runtimecost.If thecondition numberofthematrixishigh,apseudo-inverseshouldbeapplied; inthiscasepre-computationisonlypossiblewiththeprimal inter-polation.

3. ApplicationtoNICFDsimulations

Inordertoverifyandprovideinformationontheperformance oftheug-LUTmethod,threeCFDtestcasesofincreasingcomplexity levelarediscussed.Theselectedtestcasesfeatureexpansions char-acterizedbyrelevantnon-idealcompressiblefloweffects,requiring theuseofcomplexequationsofstatetoaccuratelycomputethe flowbehavior.ThesimulationsareperformedwithSU2[16],acode previouslyverifiedfornon-idealcompressiblefluiddynamics sim-ulations[24].Foralltestcases,theconvectivefluxesarediscretized byageneralizedRoescheme[25,24],andsecond-orderaccuracyis achievedwiththeMUSCLapproach[26].Ref.[16]providesamore detaileddescriptionoftheflowsolverandtheassociatednumerical methods.

Thermodynamicpropertiesneededbytheﬂowsolver,whose valueisinterpolatedfromthevaluesstoredintheLuT,are ,P,T,c,e,h,s,

∂

P

∂

e ,

∂

P

∂

e

.

Thepartialderivativesofthepressurearenecessarytocalculate theconvectiveﬂuxesinnon-idealcompressibleﬂowsimulations, see,e.g.,Ref.[18].

3.1. 2Dsupersonicnozzle

Thegeometryoftheconverging–divergingsupersonicnozzle isdepictedinFig.5a,togetherwiththeMachcontourresulting fromtheEulersimulation.Thetwo-dimensionalﬂowdomainis discretizedwithapproximately15,000triangularmeshelements

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Fig.4.2Dinviscidsupersonicnozzlesimulationresults.(a)Normalizedcomputationalcostofthesimulationemployingthelook-uptablemethod(ug-LUTsim)compared tothatofthesimulationobtainedusingtheSpan-Wagnerthermodynamicmodel(SWsim).(b)Relativemeansquareerror(RMSE)oftheMachﬁeldfromtheSWsimand ug-LUTsim.TheSWsimresultsaretakenasreference.

Fig.5. 2Dinviscidsupersonicnozzlesimulationresults,obtainedwithathermodynamicmeshofapproximately20,000nodes.(a)Machnumbercontourobtainedwiththe ug-LUTalgorithm.(b)MachnumberdistributionatcenterlineobtainedfromtheIGsim,SWsim,andug-LUTsim.

Table1

Inputparametersforthe2Dinviscidsupersonicnozzlesimulation.

Parameter Value Unit

Workingﬂuid MM –

Totalinlettemperature 530.28 K

Total-to-staticpressureratio 1.84 – Inletcompressibilityfactor 0.64 –

Inletturbulenceintensity 0.05 –

andtheworkingﬂuidissiloxane MM.Table1reportsthemain simulationparameters.

Inordertoassesstheperformanceofthemethod,the computa-tionalcostofthenozzlesimulationrelyingontheug-LUTalgorithm fortheevaluationofﬂuidproperties(ug-LUTsim)iscomparedwith oneinwhichthepropertiesareprovidedtotheﬂowsolverbythe externalthermodynamiclibrarybasedontheSpan-Wagner EoS (SWsim),foranincreasingnumberofthermodynamicmeshnodes.

Fig.4ashowsthecomputationalcostofug-LUTsimasafunctionof themeshnodes,normalizedwiththecomputationalcostofSWsim.

Fig.4breportstheroot-mean-squareerror(RMSE)betweenthe flowfieldMachnumberofug-LUTsimwithrespecttoSWsimfor dif-ferentthermodynamicmeshrefinements.Asexpected,theRMSE valuedecreaseswiththe number ofthermodynamic mesh ele-ments,whereasthecomputationalcostshowstheoppositetrend.

Fig.5b shows a comparison of theMachnumber calculated atthenozzlecenterlinewithug-LUTsimandsimulation results inwhichﬂuidpropertiesareevaluatedwiththeideal-gasmodel (IGsim)andwithamodelbasedontheSpan-WagnerEoS(SWsim). ThestreamwiseMachdistributionobtainedwithug-LUTsimiswell inagreementwiththeresultsofSWsim.Asexpected,bothdeviate fromthedistributionobtainedwithIGsim,consideringthattheinlet compressibilityfactorsigniﬁcantlydepartsfromunity(Z=0.64).

Table2

Inputparametersforthe2Dturbulentturbinecascadesimulation.

Workingﬂuid MDM –

3.2. Turbulenttransonic2Dturbinecascade

TheturbinecascadereportedinFig.6awassimulatedunder transonicconditions,inordertoevaluatetheperformanceofthe ug-LUTmethod incase ofRANS simulations.A hybridmesh of approximately40,000elementswasusedtodiscretizethe com-putationaldomain,withabout15,000quadsintheproximityof thebladesurfacetoensurey+_≈_1._The_simulation_parameters_are

listedinTable2.ForthistestcaseMDMisconsideredasworking ﬂuid.

Similarlytotheprevioustestcase,Fig.7reportsthesametrend intermsofcomputationalcostandRMSEbasedontheMachflow field.However,thecomputationalgainprovidedintheRANS simu-lationisapproximately4timeshigherascomparedtotheinviscid testcase.Fig.6bshowsthecomparisonbetweenthe dimension-less staticpressure,along theblade profile,fromtheIGsim, the SWsim,andtheug-LUTsim.TheresultsachievedwiththeSWsim arewellinagreementwithug-LUTsim,whilebothdifferfromthe onesobtainedwithIGsim.

Inordertofurtherinvestigatetheperformanceoftheproposed tabularmethod,thesameturbineconfigurationissimulatedwith thebinarymixtureMDM(85%)/MM(15%)asworkingfluid.Forthis testcase,thecostreductionisapproximatively5timeshigherthan thecomputationofthesingle-componentworkingfluid(Fig.8b), whileretainingthesameaccuracy(Fig.6a).Theseresultsshowthat

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Fig.6. 2Dturbinecascadesimulationresults,obtainedwithapproximately15,000thermodynamicmeshnodes.(a)Machnumberdistributionwiththeug-LUTalgorithm. (b)BladepressuredistributionobtainedfromtheIGsim,SWsimandug-LUTsim.

Fig.7.2Dturbinecascadesimulationresults.(a)Normalizedcomputationalcostofthesimulationemployingthelook-uptablemethod(ug-LUTsim)comparedtothatofthe simulationobtainedusingtheSpan-Wagnerthermodynamicmodel(SWsim).(b)Relativemeansquareerror(RMSE)oftheMachﬁeldfromtheSWsimandug-LUTsim.The SWsimresultsaretakenasreference.

Fig.8.2DturbinecascadesimulationresultsoperatingwiththeMDM(85%)/MM(15%)mixture.(a)Bladepressuredistributionobtainedfromug-LUTsimandSWsim.(b) Comparisonbetweencomputationaltimeassociatedwiththesingle-componentworkingﬂuid(MDM)andtheMDM(85%)/MM(15%)mixture.Resultsarenormalizedusing thecomputationaltimeassociatedwiththeSWsim.

theuseoftheLuTmethodisevenmoreattractivewhenappliedto ﬂowproblemsinvolvingmixtures.

3.3. Turbulent3Dsupersonicturbinecascade

Thesupersonic stator of the Organic Rankine Cycle turbine, documentedinRef.[27],isﬁnallyconsideredtoinvestigatethe computationalefﬁciencyoftheunstructured-basedlook-uptable methodfor three-dimensionalRANSsimulations.Thenumerical parametersofthetestcaseareprovidedinTable3.Thephysical meshconsistsofaboutonemillioncellsandthermodynamicgrid iscomposedbyapproximately20,000nodes.

Fig.9ashowsthecontourofthedensitygradient:acomplexﬂow patternofbothshock-wavesandfansispresentinthesemi-bladed region,duetothepost-expansionphenomena.Fig.9bdisplaysthe

Table3

Inputparametersforthe3DturbulentORCHIDturbinecascadesimulation.

Workingﬂuid MM –

Inletturbulenceintensity 0.05 –

densityﬁeldrelativeerrorbetweentheug-LUTsimandtheSWsim. Thedeviationintheorderof0.1%pointsoutthat,evenwitha rela-tivelycoarsethermodynamicgrid,thetabularapproachisaccurate forthree-dimensionalproblemsinvolvingcomplexﬂow phenom-ena. Furthermore, thecomputationalcost reduction for the3D

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Fig.9. 3Dturbinecascadesimulationresults.(a)Normalizeddensitygradientdistributionobtainedwiththeug-LUTalgorithm.(b)Densityrelativeerrorbetweenthe ug-LUTsimandtheSWsimresults,forapproximately15,000thermodynamicmeshnodes.

Fig.10.Summaryofthecomputationaltimefortheselectedtest-cases.Resultsare normalizedusingthecomputationaltimeassociatedwiththeSWsimandbasedon athermodynamicmeshofapproximately15,000nodes.

RANSsimulationisverysimilartotheanalysed2DRANStestcase, asshowninFig.10.

4. Performanceandmemoryassessment

Anequallyspacedstructuredgrid-basedLuTmethodwas imple-mentedwithinSU2,inordertoassesstheperformanceofug-LUT. Thestructured-basedLuTalgorithmwasconsideredforcarrying out a comparison in terms of computational costand memory requirementsbecauseofitssimpledatastructureandefﬁciency. Theturbinecascade,describedinSection3.2,isselectedas refer-encetestcasetoperformthisanalysis.Thenomenclaturesg-LUT andug-LUTisusedhereinaftertorefertothestructured-basedand theunstructured-basedLUTmethod,respectively.

4.1. Structured-gridLuTalgorithm|sg-LUT

ThestructuredgridLuT(sg-LUT)implementationfeaturesthe sameinterpolationmethodofug-LUT.Thermodynamicquery vec-tors,fortheCFDapplicationconsidered,are:(P,T),(P,),(P,s),(, T),(h,s).Byselectingthethermodynamicmeshasafunctionof pressureanddensity(Fig.11),thesearchingalgorithmisbasedon simplebinarysearchfor(P,T),(P,),(P,s),(,T),becausetheyhave atleastonecommoninputwithrespecttothermodynamicmesh. Akd-treeisusedforthe(h,s)pair.Fig.11showsanexampleof bothstructuredandunstructuredmeshofthermodynamicregion ofinterest,generatedusingthesamenumberofmeshnodes.

4.2. Comparisonsg-LUTvs.ug-LUT

Fig.12aandbreportsthenormalizedCPUtimeassociatedwith thesg-LUTsimandug-LUTsim.ThetotaltimeincludesboththeLuT pre-processingandthetotalCFDsolveriterationtime.Ascanbe noticed(Fig.12b),thepreprocessingtimebecomesarelevant frac-tionofthetotaltimeforug-LUTasopposedtosg-LUT.Thisisdue tothetrapezoidalmapgenerationforeachthermodynamicinput pair.Thisoperationisdonejustonceforsg-LUT,whencreatingthe kd-treerelativetothe(h,s)inputpair.

Theratiobetweenthetotalsimulationtimeobtainedbyug-LUT andsg-LUT(Fig.13a)indicatesthatug-LUTisfasterthansg-LUT, forthermodynamicmeshesthatareapproximatelycomposedbya numberofmeshnodeslowerthan10,000.Thesg-LUTisabout5% fasterthanug-LUTfor25,000thermodynamicmeshnodes.The ug-LUTperformanceisinagreementwiththecomputationalcostof LuTbasedonquad-treedatastructures,whosecomputationalcost hasbeenfoundtobe10%higherthanequallyspacedstructured tab-ulationmethods[13].Fig.13bshowsthecomparisonbetweenthe memoryrequirementsofug-LUTandsg-LUT.Theug-LUTmemory

Fig.11.Meshofthermodynamicregionofinterestfortheturbinecascadetestcase,asafunctionofthereduceddensityrandreducedpressurePr.(a)Exampleofstructured meshusedforsg-LUT.(b)Exampleofunstructuredmeshusedforug-LUT.

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algorithmisusedasreferencevalue.(a)Computationaltimeofthesg-LUTalgorithm.(b)Computationaltimeoftheug-LUTalgorithm.

Fig.13.TotalCPUtimeandmemoryratiobetweenug-LUTandsg-LUT.(a)Computationaltimeratio.(b)Memoryrequirementsratio.

Fig.14.Relativemeansquareerrorratiobetweensg-LUTandug-LUT.(a)RMSEratiooftheconservativevariables.(b)RMSEratioofpressureandtemperature.

requirementsare higherthan sg-LUT mainlyduetothe follow-ingreasons:(i)inug-LUTthetrapezodialmapsarecreatedforall thermodynamicinputpairs;(ii)forug-LUTtheunstructured-mesh connectivityhastobestored.Forpracticalapplications,however, sincethermodynamicmeshesfeaturingaround10,000elements aredeemedsufﬁcientfortherequiredlevelofaccuracy,the abso-lutememoryassociatedneverexceeded200Mb.Furthermore,for problems discretized onlarge domains both the preprocessing computationalcostandthememoryburdenareexpectedtobea negligiblefractionwhencomparedwiththeCPUtimeandmemory requirementsoftheCFDsimulation.

Finally,acomparativeassessmentoftheRMSEwithrespectto theSWsimiscarriedoutforbothug-LUTandsg-LUT.Fig.14depicts theratiobetweentheRMSEobtainedbyug-LUTandsg-LUTrelative totheconservativevariables,

v

1,

v

2,e(Fig.14a),pressureand

temperature(Fig.14b).TheRMSEiscalculated,asshowninSection

3,withrespecttoSWsim.Theug-LUTalgorithmismoreaccurate thansg-LUT, fortheﬂuxes

v

1,

v

2,e,andthepressure while

theoppositeoccurswithregardtothedensityandtemperature. Withoutbeingexhaustive,theseresultsindicatethatunstructured tabularmethodsmaybeadvantageouswhenitcomestoaccuracy

ascomparedtostructuredgridscharacterizedbythesamenumber ofnodesandlevelofreﬁnement.

5. Conclusions

This paper documents the ug-LUT method, a novellook-up table method that can be used to improve the computational performance in non-ideal compressible ﬂuid dynamics (NICFD) simulations. The methodis based onan unstructured mesh in combination witha trapezoidal-map searchingalgorithm and a piece-wiseinterpolationmethodbasedonthedualityapproach. Thealgorithmwassuccessfullyimplementedintheopen-source codeSU2anditsperformanceassessedinthreeparadigmaticNICFD cases:(1)aninviscid2Dsupersonicnozzle;(2)a2DRANS tran-sonicturbinecascade;(3)a3DRANSsupersonicstatorrow.Inall cases,thermodynamicpropertymodelofreferenceisbasedona multi-parametertechnicalequationofstate.

Theoutcomeofthisstudycanbesummarizedasfollows: • The ug-LUT methodprovides a computationalcost reduction

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directlybymeansofanexternalﬂuidpropertylibraryof approx-imatelyoneorderofmagnitudeforRANScomputations,whereas itistwicelessexpensiveincaseofinviscidsimulations. • Theaccuracylevelwasfoundtobesatisfactory(RMSE<0.01%)for

engineeringapplications,withasimplelinearinterpolationand arelativelycoarsethermodynamicgrid(oftheorderof15,000 elements).

• Themethodisveryefficientforflowsimulationsinvolving mix-tures as working fluids, for which direct calculation of fluid propertiesmightbeprohibitive.Therelativecomputationalgain, fortheanalyzedtestcase, isfivetimeshigherifcomparedto simulationsinvolvingpurefluids.

• ug-LUTcanberegardedasanalternativealgorithmtostructured LuTmethods,providingthepossibilityofusingmeshreﬁnement andfeaturingcomparableperformanceandaccuracy.

Currentworkisfocusedonextendingtheug-LUTmethodto enableautomaticdifferentiationoftheLuTforadjoint-basedshape optimization of NICFD problems and to allow its use in other demandingsimulations,likedynamicsystemsimulationsofenergy conversionsystems.

Acknowledgements

ThisresearchhasbeensupportedbyRobertBoschGmbHand theAppliedandEngineeringSciencesDomain(TTW)oftheDutch OrganizationforScientiﬁcResearch(NWO),TechnologyProgram oftheMinistryofEconomicAffairs,grantnumber13385.

AppendixA.

Here,thepseudo-codesofthealgorithmsusedforthe Trape-zoidalMappresentedinSec.2.2arereported.

Algorithm1. BuildTrapezoidalMapis.

1:input:listofuniqueedges:Edges;

2:listofedgetofaceconnectivities:EdgeToFace; 3:listofthex-coordinatesoftheedges:xsamples; 4:listofthey-coordinatesoftheedges:ysamples;

5:output:listofuniquexbandsthroughwhichtosearch:XBands; 6:listofconnectivityofedgestoagivenuniquex-band(orderbyy-valueof edgeatmiddleofband):YWithinBands;

7:FilteroutverticaledgesfromXsamplescreatinguniquelist:xBands; 8:foreachbandbinXBandsdo

9: foreachedgeeinEdgesdo

10:ifeintersectsbthenAddetoYWithinBands 11:endif

12: endfor

13: SorttheedgesinYWithinBandsaccordingtotheyvalueoftheedgeinthe centeroftheband

14:endfor

15:return:XBands,YWithinBands;

Algorithm2. TrapezoidalMapis.

1:input:2Dquerypoint(whattosearchfor):x,y; 2:listofedgetofaceconnectivities:EdgeToFace; 3:listofuniquexbandsthroughwhichtosearch:XBands;

4:listofconnectivityofedgestoagivenuniquex-band(orderedbyy-valueof edgeatmiddleofband):YWithinBands;

5:output:theindexofthefaceinwhichthequerypointlies:CF; 6:

7:UsebinarysearchtoﬁndbandbinXBandswhichcontainsx;

8:Withinbandbuseabinarysearchtoﬁndedgee1(belowthequerypoint) ande2(abovethequerypoint);

9:Useinterpolationtocheckifedgeisbeloworabovethequerypoint; 10:ThecontainingfaceCFisgivenbytheintersectionoftheedgetoface connectivitiesofe1ande2;

11:return:CF;

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