Computational and Experimental Characterization of the Mach 6 Facility Nozzle Flow for the Enhanced Injection and Mixing Project at NASA Langley Research Center TomaszG.Drozda,† KarenF.Cabell,† BradleyJ.Passe,‡ andRobertA.Baurle,§ NASALangleyResearchCenter,Hampton,VA,23681 Computational fluid dynamics analyses and experimental data are presented for the Mach 6 facilitynozzleusedintheArc-HeatedScramjetTestFacilityfortheEnhancedInjectionandMixing Project(EIMP).Thisproject,conductedattheNASALangleyResearchCenter,aimstoinvestigate supersoniccombustionramjet(scramjet)fuelinjectionandmixingphysicsrelevanttoflightMach numbers greater than 8. The EIMP experiments use a two-dimensional Mach 6 facility nozzle to provide the high-speed air simulating the combustor entrance flow of a scramjet engine. Of interest are the physical extent and the thermodynamic properties of the core flow at the nozzle exit plane. The detailed characterization of this flow is obtained from three-dimensional, viscous, Reynolds-averagedsimulations. Thermodynamicnonequilibriumeffectsarealsoinvestigated. The simulationsarecomparedwiththeavailableexperimentaldata,whichincludeswallstaticpressures as well as in-stream static pressure, pitot pressure and total temperature obtained via in-stream probespositionedjustdownstreamofthenozzleexitplane. I. Introduction THE Arc-Heated Scramjet Test Facility (AHSTF)1–3 is a high-enthalpy wind tunnel used for testing supersonic combustionramjet(scramjet)engines. Withrecentupgradestoextendthelow-Machsimulationcapability, the facility can produce total enthalpies corresponding to flight Mach numbers of about 2.5 to 8. The high total en- thalpies are achieved by over-heating a portion of the total facility air with an electric arc and then mixing it with unheatedbypassairtoobtainthedesiredmixturetotalenthalpy. Thecombinedairmixtureisthenexpandedthrough a converging-diverging facility nozzle. Several facility nozzles with various exit Mach numbers are available with the nozzle selection depending on the desired flight simulation. The nozzle Mach number can be equal to the flight Machnumberforafree-jetconfiguration,orlessthantheflightMachnumberfora“semi-freejet”or“direct-connect” configuration,wheresomelevelofprecompressionissimulated. Theflowprofilesattheexitofthefacilitynozzledependonthenozzlegeometry4 (e.g.,axisymmetric,squareor rectangular), as well as, to a lesser degree, the heat transfer to the nozzle walls. Furthermore, the surface contour of a facility nozzle is carefully designed for operation at a specific value of the total enthalpy and Mach number to preventtheformationofshockorexpansionwavesduringtheflowexpansionprocess. However, becausedesigning andfabricatingafacilitynozzleforeachconditionofinterestisnotalwaysfinanciallypossible,theexistingnozzles areoftenusedatoff-designconditions,ormorespecificallyattotalenthalpiesthatdifferfromthatforwhichthenozzle wasdesigned. Inthosesituations,itisprudenttocharacterizetheflowprofilebecauseboththephysicalextentandthe aero-andthermodynamicpropertiesofthenozzlecoreflowwillchangeasafunctionofthetotalenthalpy. TheEnhancedInjectionandMixingProject(EIMP)5 beingconductedattheNASALangleyResearchCenteris utilizestheAHSTFforexperimentalinvestigationsofscramjetfuelinjectionandmixingprocesses. TheEIMPaims toimproveunderstandingofthefundamentalphysicsthatgoverntheseprocesses,aswellas,todevelopconceptsand functionalrelationshipsbetweenmixingperformanceandlossesrelevanttoflightMachnumbersgreaterthan8. The †ResearchAerospaceEngineer,HypersonicAirbreathingPropulsionBranch,AIAASeniorMember. ‡ResearchEngineer,AnalyticalMechanicsAssociates,Inc.,AIAAMember. §ResearchAerospaceEngineer,HypersonicAirbreathingPropulsionBranch,AIAAAssociateFellow. 1of27 AmericanInstituteofAeronauticsandAstronautics researchapproachisaclosely-coupledexperimentalandcomputationaleffort,withexperimentsintendedforexploring injectionconceptsandforanchoringcomputationalfluiddynamics(CFD)simulations.TheexperimentsutilizeaMach 6facilitynozzle,designedforMach7totalenthalpy,toproducethehigh-speedairsimulatingthecombustorentrance flow of a scramjet engine. Although the combustor entrance Mach number matches that for flight, the maximum totalenthalpyfortheseexperimentsislimitedbythethermal-structurallimitoftheuncooledexperimentalhardware, and corresponds to a Mach 4.25 flight enthalpy. Because this total enthalpy is significantly lower than the Mach 7 designtotalenthalpy,someuncancelledwavesandreducedexit“core”sizemightbeexpected. Additionally,detailed knowledge of the nozzle exit flow is necessary for accurate CFD simulations of the mixing flow fields. Therefore, computationalsimulationsoftheMach6nozzleflowattheplannedEIMPtestconditionswereperformedandnozzle exitflowsurveytestswereconductedtogatherdatatovalidatetheCFD.Thecomputationalsimulationsincludethe effects of thermal nonequilibrium, which can be present as a result of rapidly expanding the heated air containing rotationally and vibrationally excited species through a facility nozzle.6 It should be noted that previously reported pretest CFD simulations7 utilized the as-designed nozzle geometry, whereas the current work utilizes the as-built geometry, with the most significant difference being that the as-built throat area is 7% smaller than the as-designed nozzlethroatarea. The paper is organized as follows: first, a description of the Mach 6 nozzle flow survey experiment is provided. Thisincludesthedescriptionsofthefacilityandnozzle,andthetestconditionsandacquireddata. Next,anoverview of the CFD approach is given. Discussion of numerical considerations that apply to all simulations is presented, as well as, results illustrating the impact of several numerical modeling choices (grid resolution, turbulence model, thermalboundarycondition,andthermalnonequilibrium)onthecoreflow. TheCFDcalculationsarethencompared with experimentally measured wall static pressures and in-stream nozzle-exit measurements of static pressure, pitot pressureandtotaltemperature. Thepaperisthensummarized,andfutureeffortsaredescribed. II. Mach6NozzleFlowSurveyTests As described above, in the AHSTF, a portion of the total facility air is heated by an electric arc and then mixed with unheated air to obtain the desired mixture total enthalpy. The arc-heated “main” air exits the heater at a bulk temperatureofabout4000◦Randentersaplenumchamber,upstreamofthenozzle,wheretheunheated“bypass”air isintroduced. Thebypassairisinjectedradiallytobreakuptheswirlingmotionofthemainairandpromotemixing ofthetwostreams. Thismethodofheatingonlyaportionofthetotalairtoelevatedtemperaturesandthenmixingit withcoolerairtoachievethedesiredtotalenthalpyisnecessarybecausethearcheatercannotoperateinastablemode whileprocessingthefullfacilityairflowrates. Asaresultofarc-heating,theaircontainstraceamountsofnitricoxide (NO)asreportedbyCabellandRock8. Consideringthiswork,themolefractionofNOinthetestgasfortheEIMP testconditionsisestimatedtobeabout1%. Flowintheplenumchamberissubsonicandnearlystagnated,therefore,thestagnationpressureiswellapproxi- matedbyaplenumwallpressuremeasurement. Totalmassflowrateismeasuredusingcalibratedsonicventuriflow meters in both the main and bypass air supply lines. Plenum stagnation enthalpy is calculated using a sonic throat enthalpycalculation,whichaccountsforcaloricimperfections.9Inputstothiscalculationarethemeasuredstagnation pressure,themeasuredtotalmassflowrate,themeasurednozzlethroatarea,andanassumednozzledischargecoef- ficient of one. This last assumption can be readily verified using the nozzle CFD, which reveals that for the EIMP testconditions,theerrorassociatedwiththisassumptionisatmost0.7%. Theplenumstagnationtemperatureisthen calculatedfromthestagnationenthalpyassumingthermallyperfectair. Thefacilitynozzleusedinthecurrentexperiments,referredtoasthe“Mach62D”nozzle,wasdesignedtoproduce a Mach 6 core flow with minimal nonuniformities at a Mach 7 flight enthalpy.4 A side view of the nozzle is shown in Fig. 1. The top and bottom walls are contoured, but the side walls are flat. The flow cross-section is 10 inches widethroughoutwithaheightof0.1,and10inchesatthethroatandexit, respectively. Thelengthoftheexpansion section from the throat to the exit plane is 70 inches. The nozzle throat section is made of copper, which is water cooledtoallowforcontinuousruntimes. Theexpansionsection,startingfromabout12.5inchesdownstreamofthe throat, is made from stainless steel. The nominal operating conditions used in the EIMP correspond to a plenum stagnation pressure of 625 psia and a plenum stagnation temperature ranging from 1315 to 1760◦R. The range is determined by the condensation limit of expanding air on the low end, and by the thermal-structural limit of the uncooledexperimentalhardwarelocateddownstreamofthenozzleonthehighend. ThesevaluescorrespondtoMach 3.4 and 4.25 flight enthalpies, respectively, at a flight dynamic pressure of 1500 psf. The nominal, ideal expansion characteristics,basedonthegeometricarearatio,forthenozzlearesummarizedinTable1. 2of27 AmericanInstituteofAeronauticsandAstronautics Figure1.Cross-sectionoftheMach6facilitynozzle(dimensionsininches). AsthefacilityairexpandsthroughthethroatandintothesupersonicportionofthelongMach6nozzle,fairlythick (uptoabout2inches)boundarylayersdevelopalongitswalls.7Theseboundarylayersreducethenozzleeffectiveexit area,andtherebychangethenozzlearearatioandthecoreflowexitflowproperties. PreviouslyreportedpretestCFD simulations7 indicated that the resulting core flow Mach number is about 6.5. Furthermore, the turbulent boundary layers exhibit a reduction in the growth rate, under the favorable pressure gradient flow conditions as compared to the zero-pressure gradient flat plate case. This phenomena is important because it impacts the final thickness of the boundarylayerandtherefore,thenozzlecoreflowexitproperties. Sincemost,ifnotall,oftheturbulencemodelsare calibratedusingthezero-pressure-gradientflatplateexperimentaldata,10 thepredictionsunderadverseorfavorable pressuregradientsmaynotbeoptimal. Furthermore,sinceCFDisoftenusedfornozzledesign,anymispredictionof the boundary layer character during the nozzle design phase has the potential to impact the actual Mach number of thenozzle,especiallyforhighMachnumbernozzles,whicharegenerallymuchlongerthantheirlowMachnumber counterparts,andtherefore,allowtheboundarylayerstodevelopandgrowoverlongerdistances. Allfourwallsofthenozzleareinstrumentedwithstaticpressureports. Thesidewallsaredefinedasrightandleft asviewedlookingupstream. Anaxialrowofstaticpressureportsexistsalongthecenterlineofboththebottomwall andtherightsidewallatthelocationsindicatedinFig.1. Theleftandtopwallscontainportsatasubsetofthesesame axiallocations. Additionally,atanaxialstationofx=68.45inch(1.55inchesupstreamofthenozzleexitplane),there Table1.Nominalplenum,andinviscidandisentropicexpansionconditionsfortheMach6facilitynozzle. Property LowTot. Temp. HighTot. Temp. TotalPressure(psia) 625 625 TotalTemperature(◦R) 1315 1760 Mach 6.93 6.87 m˙ (lbm/s) 8.92 7.66 Density(lbm/ft3x10−3) 3.345 2.459 Pressure(psia) 0.1574 0.1608 Temperature(◦R) 126.6 175.9 U(ft/s) 3828 4473 Re/L(1/in) 238e3 160e3 AreaRatio 102.04 102.04 3of27 AmericanInstituteofAeronauticsandAstronautics Figure2.ACADimageandaphotographofthein-streamrakeassembly.Fromleft-to-right:totaltemperature,pitotpressure,andstatic pressurerakes. arefiveportsoneachwall,distributedspanwise(onthetopandbottomwalls)orvertically(onthesidewalls). Measurements in the core flow of the nozzle exit were obtained by an in-stream rake system consisting of three rakes mounted on a pallet: a total temperature (T ) rake, a pitot pressure (P ) rake, and a static pressure (P ) rake. t t2 s An isometric view of the rake assembly, as well as, a photograph are shown in Fig. 2. Spanwise spacing between rakes is fixed at two inches. All probe tips lie in a common axial plane, which is located 0.050 inches downstream of the nozzle exit plane. The rake assembly was translated to three different spanwise positions to provide data for comparisonswiththeCFDsimulations. Vertical spacing between the probes for each rake is shown in Fig. 3. The pitot pressure probes are constant diameter tubes having 0.125-inch outer diameter and 0.035-inch wall thickness. The stagnation temperature probes are 0.125-inch outer diameter tubes having 0.010-inch wall thickness and an exposed thermocouple bead recessed 0.25 inches inside the tube. They are the same design used in previous flow surveys of other AHSTF nozzles,11 with the exception that the tube material and thermocouple type (type K) were selected for the lower temperature range associated with the EIMP conditions. The static pressure probe geometry, shown in Fig. 4, is based on the design by Pinckney12. However, this particular probe geometry was designed for a lower freestream Mach number and intended for flow surveys of the Mach 2.2 and 3.5 facility nozzles. Probe surface pressure distributions from axisymmetricCFDsimulationsassumingthermallyequilibratedflowovertheprobeatfreestreamMachnumbersof 2.2, 3.5 and 6.35 are shown in Fig. 5. These results are used to derive a probe recovery factor, which is a ratio of measuredtofreestreamstaticpressure,asafunctionoffreestreamMachnumbertobeappliedtothemeasurements. The core flow conditions for the Mach 6.35 flow for the static pressure probe simulations were obtained from the previous Mach 6 nozzle simulations of Drozda et al.7 utilizing the as-designed nozzle geometry and the high total temperatureconditions. Thevaluesoftherecoveryfactorsforthecurrentexperimentsare1.29forthecore,and1.08 forthetop-andbottom-moststaticpressureprobes. Alowervalueisusedforthelatter,becausetheseprobesretrieve measurementsfromthetopandbottomnozzlewallboundarylayers,wheretheMachnumberisestimatedtobeabout 4.7. Additional simulations showed that the recovery factor for the core flow would be about 3% lower for the low thanhightotaltemperatureconditions,andabout1%lowerifthermalnonequilibriumeffectswereconsidered. All nozzle wall and in-stream pressure data were acquired at 12 Hz with an electronic scanning pressure (ESP) system. Allotherexperimentaldatawereacquiredat50Hz. Themeasurementuncertaintyforthenozzlewallpres- suresupstreamofthex=68.45inchesstationandforthepitotpressuresis±0.015psi. Themeasurementuncertainty forthenozzlewallpressuresatthex=68.45-inchstationandforthein-streamprobestaticpressuresis±0.0075psi. Themeasurementuncertaintyfortheprobetotaltemperaturesisestimatedtobe±4.7◦R.Alldataweretime-averaged over a six second period of “steady” conditions. The time-averaged values of facility stagnation pressures and tem- peraturecanvaryrun-to-runfromtheirtargetnominalvalues(listedinTable1). Thesevariationscanbeuptoabout 2%,and3%forthestagnationpressureandstagnationtemperature,respectively. Forthepurposeofcomparisonofthe 4of27 AmericanInstituteofAeronauticsandAstronautics Figure3. Sideviewsofthestaticandpitotpressurerakes,andthetotaltemperaturerakeshowingverticalprobespacing(dimensionsin inches). Figure4.Staticpressureprobegeometry(dimensionsininches). 5of27 AmericanInstituteofAeronauticsandAstronautics Figure5.StaticpressureprobesurfacedistributionforfreestreamMachnumbersof2.2,3.5and6.35 6of27 AmericanInstituteofAeronauticsandAstronautics experimentaldatatotheCFD,allthetime-averagedwallandprobepressuredataarerescaledbyaratioofthetarget andtime-averagedfacilitystagnationpressureandallthetime-averagedprobetotaltemperaturedataarerescaledbya ratioofthetargetandtime-averagedfacilitystagnationtemperatures. Additionally,arecoveryfactorisappliedtothe staticpressureprobemeasurement,asdescribedabove,dependingontheprobelocation. UsingtheresultsoftheCFD simulationsdiscussednext,thepredictedMachnumberinthecoreflowwasusedtocalculatetherecoveryfactor. No corrections were applied to the total temperature probe measurements to account for radiation heat loss because the previous nozzle flow surveys using this probe design11 demonstrated a recovery factor of one for even higher total enthalpyflowconditions(i.e.,Mach5). III. NumericalSimulationsandSensitivities The numerical simulations were performed using the Viscous Upwind aLgorithm for Complex flow ANalysis (VULCAN-CFD) code.13 VULCAN-CFD is a multi-block; hybrid structured- and unstructured-grid, cell-centered, finite-volumesolverwidelyusedforall-speedflowsimulations. Forthiswork,Reynolds-averagedsimulations(RAS) wereperformed. TheadvectivetermswerecomputedusingaMUSCLscheme14withtheLow-DissipationFlux-Split Scheme(LDFSS)ofEdwards15. Anonreacting, thermallyperfectmixtureof21%oxygen(O ), 78%nitrogen(N ), 2 2 and1%NOwasusedforthefacilityair. AsmallamountofNOwaspresenttoaccountforproductionofthisspecies inthearc-heateroftheexperimentalfacility.8 Thethermodynamicpropertiesofmixturecomponentswerecomputed usingthecurvefitsofMcBrideetal.16. Forcalculationsutilizingthethermodynamicnonequilibriummodel,17 these curve fits are evaluated at the vibrational temperature.6 The governing equations were integrated using an implicit diagonalized approximate factorization (DAF) method18 and, for some cases, incomplete LU (ILU) factorization.19 ThecurrentworkusedthebaselineturbulencemodelofMenter20,althoughsomeresultsutilizinganexplicitalgebraic Reynolds-stress model (EARSM) of Rumsey and Gatski21 are also shown to assess turbulence model sensitivity. The Reynolds heat and species mass fluxes were modeled using a gradient diffusion model with turbulent Prandtl and Schmidt numbers of 0.9 and 0.5, respectively. The Wilcox wall matching functions10 were also used, where appropriate, to reduce the near-wall grid resolution requirements. However, integrate-to-the wall simulations were alsoperformedtoassesstheimpactofnear-wallmodelingontheboundarylayerthickness. Agridsensitivitystudy wasperformedforseveralsimulationswithnegligiblechangesinflowpropertiesbeingcomputedwhenthefinegrid resolutionwasreducedbyafactoroftwoineachdirection. Allsimulationswereconvergeduntilthetotalintegrated mass flow rate and the total integrated heat flux on the walls remained constant to about six significant digits. This typically occurred when the value of the L -norm of the steady-state equation-set residual decreased by about 4-5 2 ordersofmagnitude. Subsonicinflowboundaryconditionsspecifyingatotalpressureof625psiaandtemperatureof1760◦Rwereused attheinflowlocatedupstreamofthenozzlethroatforthehighenthalpycase.Forthelowenthalpycase,atotalpressure of625psiaandtemperatureof1315◦Rwereused. First-orderextrapolationwasusedatthesupersonicoutflowofthe nozzle. The walls were modeled using a no-slip boundary condition for the velocities. The temperatures at the nozzlewallswereobtainedbylocallysolvingaone-dimensional(1D),steady-stateheatconductionequationwiththe experimentally-obtainedaveragevaluesofthenozzleouter-walltemperaturesasaninput. However,isothermalwall temperatureboundaryconditionswerealsoincludedinthisstudytoevaluatethesensitivityoftheboundarylayerto the near-wall thermal environment. The pitot pressure at the nozzle exit plane was computed iteratively using the Rankine-Hugoniotjumpconditionswiththeflowassumedtobevibrationallyfrozenthroughthenormalshockjump (for simulations that included thermal nonequilibrium modeling) and then thermally equilibrated to stagnation. The simulationswereinitializedusingasolutionofthecorresponding1Dinviscid, isentropicandadiabaticsolutiontoa converging-divergingnozzleproblem. Taking advantage of the nozzle symmetries, only the top-left quarter of the full nozzle geometry was simulated. Thex,y,andz,coordinatesarealignedinthestreamwise,cross-stream(pointingtowardthenozzlecurvedwall),and spanwise(pointingtowardthenozzlesidewall)directions,respectively,withtheoriginlocatedatthenozzlecenterline atthethroat. Allofthespatialdimensionsareininches. Thegridusedwiththewallmatchingfunctions(WMF)has about3.9millioncellsandwillbereferredtoastheWMFgrid. ItshouldbenotedthatallsimulationsontheWMF gridalsoutilizedturbulencemodelswiththewallmatchingfunctions. Theintegrate-to-the-wall(ITW)gridhasabout 34.5 million cells, does not require the use of the wall matching functions, and will be referred to as the ITW grid. The grid densities near the centerline of the nozzle are similar for both grids in the cross-flow directions, with the ITW grid containingabout twice as many points inthe streamwise direction. The resulting values of dimensionless walldistance(y+)ofthefirstoff-the-wallcell-center,obtainedforthethermallyperfectsimulationsatthehighlimit 7of27 AmericanInstituteofAeronauticsandAstronautics Figure6.Contoursofthedimensionlesswalldistance(y+),obtainedforthethermallyperfectsimulationsatthehighlimitoftotaltemper- ature(left)withwallmatchingfunctions(WMF),and(right)integrate-to-the-wall(ITW).Theinsetshowsaclose-upofthenozzlethroat section.Top-leftquarterofthefacilitynozzleisshown. of the total temperature are shown in Fig. 6. The results for the cases at the low limit of the total temperature are comparable. Forsimulationsutilizingthewallfunctions,they+valuesrangedfromabout10to80,withanaverage of about 25, and the largest values located at the nozzle throat. The y+ values for the ITW grid were less than one everywhere. Figure 7 shows line plots of the Mach number, static temperature, and static and pitot pressures at the nozzlecenterlinevs. axialdistanceobtainedfromtheCFDsolutionswithWMF(lines)andITW(symbols)grid. The insetoneachfigureprovidesfocusonthedatanearthenozzleexitplane. Thechangesinthecenterlineexitvaluesas afunctionofnear-wallmodelingapproacharerelativelysmallwith0.1,0.3,and1.5and0.8%fortheMachnumber, statictemperature,andstaticandpitotpressures,respectively.Figure8showscontourplotsoftheMachnumber,static temperature, and static and pitot pressures at the nozzle exit plane obtained from the CFD solutions with the WMF and ITW grids. It should be noted that for the static pressure contours the color scale has been focused on the exit plane such that even small differences are visible. Examination of the color scale values, however, reveals that the variationofthestaticpressuresinthecoreflowiswithin2%. Overall,thedifferencesarefairlysmallbetweenthese solutions,however,thereisaslightthickeningofthethermalboundarylayersforsimulationsutilizingtheWMFgrid. Thisthickeningdoesnotseemtosignificantlyimpacttheextentofthecoreflowforthequantitiesofinterest. These simulationsindicatethat,forthecurrentcases,theWMFmodelsperformquitesatisfactorily. A grid sensitivity study was performed for simulations with the WMF grid for the thermally perfect case at the highlimitofthetotaltemperature. OnlytheWMFgridwasusedbecause,asshownabove,theITWsolutionsdonot produce significantly different results. Figure 9 shows line plots of the Mach number, static temperature, and static and pitot pressures at the nozzle centerline vs. axial distance obtained from the CFD solutions on the coarse (lines) andfine(symbols)grids. Theinsetoneachfigureprovidesfocusonthedatanearthenozzleexitplane. Thechanges in the centerline exit values as a function of grid density are 0.3, 0.5, and 2 and 1.4% for the Mach number, static temperature,andstaticandpitotpressures,respectively. Since,accordingtotheGridConvergenceIndex(GCI),22for factoroftwogridrefinements,aconservativeestimateofthenumericalaccuracyisconservativelyproportionaltothe differencebetweencoarseandfinegridsolutions,wecanexpectthatournumericalerrorsforthesolutionsonthefine gridarecomparabletothosereportedabove. Figure10showscontourplotsoftheMachnumber,statictemperature, and static and pitot pressures at the nozzle exit plane obtained from the CFD solutions on the coarse (C) and fine (F)grids. Thedifferencesareminimalbetweenthesesolutions,however,thereisasmallthickeningoftheboundary layersonthecoarsegrid. Thisthickeningisresponsibleforthereductionintheeffectiveexitareaandcorresponding reductionintheexitMachnumberforthecoarsegridsolution. The sensitivity to the turbulence model was assessed by comparing the CFD simulations utilizing the baseline turbulentphysicsmodelofMenter20 tothosewiththeexplicitalgebraicReynolds-stressmodel(EARSM)ofRumsey andGatski21 forthethermallyperfectcase,atthehighlimitofthetotaltemperature. Figure11showsthelineplotof theMachnumbervs.axialdistance,andMachnumbercontoursatthenozzleexitplanefortheCFDsolutionsobtained 8of27 AmericanInstituteofAeronauticsandAstronautics Figure7. LineplotsoftheMachnumber,statictemperature,andstaticandpitotpressuresatthenozzlecenterlinevs. axialdistance obtainedfromCFDsolutionswithwallmatchingfunctions(WMF)(line),andintegrate-to-the-wall(ITW)(symbols)grids. Theinseton eachplotprovidesafocusonthedatanearthenozzleexitplane. 9of27 AmericanInstituteofAeronauticsandAstronautics Figure8.ContourplotsoftheMachnumber,statictemperature,andstaticandpitotpressuresatthenozzleexitplaneobtainedfromCFD solutionswithwallmatchingfunctions(WMF),andintegrate-to-the-wall(ITW)grids. 10of27 AmericanInstituteofAeronauticsandAstronautics