Wall-Resolved Large-Eddy Simulation of Flow Separation Over NASA Wall-Mounted Hump AliUzun∗ National InstituteofAerospace,Hampton,VA23666 MujeebR.Malik† NASALangleyResearchCenter,Hampton,VA23681 Thispaperreportsthefindingsfromastudythatapplieswall-resolvedlarge-eddysimulationtoinvestigate flowseparationovertheNASAwall-mountedhumpgeometry. Despiteitsconceptuallysimpleflowconfigu- ration, this benchmark problem has proven to be a challenging test case for various turbulence simulation methodsthathaveattemptedtopredictflowseparationarisingfromtheadversepressuregradientontheaft regionofthehump. Themomentum-thicknessReynoldsnumberoftheincomingboundarylayerhasavalue thatisneartheupperlimitachievedbyrecentdirectnumericalsimulationandlarge-eddysimulationofin- compressibleturbulentboundarylayers. ThehighReynoldsnumberoftheproblemnecessitatesasignificant number of grid points for wall-resolved calculations. The present simulations show a significant improve- mentintheseparation-bubblelengthpredictioncomparedtoReynolds-AveragedNavier-Stokescalculations. The current simulations also provide good overall prediction of the skin-friction distribution, including the relaminarizationobservedoverthefrontportionofthehumpduetothestrongfavorablepressuregradient. Wediscussanumberofproblemsthatwereencounteredduringthecourseofthisworkandpresentpossible solutions. A systematic study regarding the effect of domain span, subgrid-scale model, tunnel back pres- sure,upstreamboundarylayerconditionsandgridrefinementisperformed.Thepredictedseparation-bubble lengthisfoundtobesensitivetothespanofthedomain. Despitethelargenumberofgridpointsusedinthe simulations, somedifferencesbetweenthepredictionsandexperimentalobservationsstillexist(particularly forReynoldsstresses)inthecaseofthewide-spansimulation,suggestingthatadditionalgridresolutionmay berequired. I. Introduction Smooth-body flow separation is an important problem of practical interest, due to its appearance in many tech- nologicalapplications. Theprobleminvolvesaboundarylayerattachedtoasolidsurface, whichbecomesdetached fromthesurfaceafterinteractionwithanadversepressuregradientgeneratedbyachangeinbodycontourorthepres- enceofashock. Thisphenomenoncommonlyoccursinflowsoveraircraftwings,helicopterrotors,turbomachinery bladesandhigh-liftconfigurations,tonameafew. Separationoftenleadstoincreasedaerodynamicdrag,stallandre- ducedsystemperformance. SuchseparatedflowsaregenerallydifficulttopredictbecausetheyinvolvehighReynolds number turbulence. These high Reynolds number flows have been traditionally studied using techniques such as Reynolds-Averaged Navier-Stokes (RANS) calculations, wall-modeled large-eddy simulations (WMLES) or hybrid RANS-LES type approaches. Inthe case ofRANS, available turbulence models commonly failtoproperly account fornon-equilibriumeffectsinseparatedflows,andthereforeleaveroomforimprovement. Anexampledemonstrating thefailureofRANSinalow-speedflowseparationproblemcanbefoundinthepaperbyRumseyetal.1 Theproblem involves a wall-mounted hump geometry, also known as the NASA hump, representative of the upper surface of an airfoil,asdepictedinFigure1(a). Theaftportionofthehumpgeneratesanadversepressuregradient,whichcauses boundary layer separation at x/c ≈ 0.665 and flow reattachment further downstream, at x/c ≈ 1.11. Figure 1(b) depictsthefailureofRANSinthepredictionofseparation-bubblelength,whichisoverestimatedbyabout35%. ThesamebenchmarkproblemhasalsobeenstudiedusingWMLES(e.g.,seethestudiesofAvdisetal.,3 Shuret al.,4 Park,5 Duda and Fares,6 Iyer and Malik7). In WMLES, the critical near-wall region of the turbulent boundary ∗SeniorResearchScientist,SeniorMemberAIAA. †SeniorAerodynamicist,ComputationalAeroSciencesBranch,MS128,FellowAIAA. 1of22 AmericanInstituteofAeronauticsandAstronautics (a) Experimentalmodel. (b) SkinfrictionpredictedbyRANS. Figure1. NASAwall-mountedhumpexperiment2. layer is modeled, rather than resolved, to avoid stringent grid resolution requirements. Despite generally promising results from these studies, the overall success of WMLES at present can be characterized as mixed at best. The success or failure of WMLES in separated flows strongly depends on the accuracy of the wall model employed in the near-wall region. While wall models have been proposed for both equilibrium and non-equilibrium conditions, the true performance of some wall models is hard to assess since certain critical information, such as surface skin- friction distribution, is not always provided or grid convergence of the solution is not clearly demonstrated. From aerodynamicconsiderations,theskin-frictiondistributionisoneofthemostimportantquantitiesandWMLESusually doesnotyieldsatisfactoryresultsforthisquantity. TherecentstudybyIyerandMalik7 showedinparticularthatfor theNASAwall-humpproblem,theskinfrictionpredictedbyaWMLESbasedonanequilibriumwallmodelneeded improvement. Forthecurrentconfiguration,thereisalsoaregionofstrongfavorablepressuregradientoverthefront partofthehump,whereRANSandWMLESbothtendtoyieldinaccurateskin-frictiondistributions. There have also been studies of the wall-mounted hump problem in which traditional LES was applied without any wall models (e.g., see the studies by You et al.8 and Franck and Colonius9). However, careful examination of thesestudiesrevealsthatthenear-wallgridresolutionsarenotsufficient. Inparticular,theboundarylayerresolution inwallunitsreportedbyYouetal.8 cannotbeattainedwiththenumberofgridpointsused(about7.5millionpoints total)andtherefore,isinerror. Aswillbediscussed,alotmoregridpointsareneededforawall-resolvedsimulation athighReynoldsnumber. Eventhoughthesepriorstudieshavereportedsomeencouragingresults,webelievethatthe practiceofperforminganLESusinganunder-resolvedgridwithoutwallmodelingisill-foundedbecausethenear-wall regionoftheboundarylayerhastobemodeledifthegridresolutionisnotsufficienttoproperlyresolvethatregion. Asurveyofthecurrentliteratureconfirmsthelackofwell-resolvedturbulencesimulationsforcomplexseparated flowsathighReynoldsnumber. Toourknowledge,fortheNASAhumpproblem,thebest-resolvedsimulationprior to the current work is the coarse-grid “DNS” performed by Postl and Fasel.10 This DNS is labeled “coarse-grid” because the resolution in wall units is more in line with a wall-resolved LES (WRLES) rather than DNS. A total of 210 million points were used in their DNS with a relatively small spanwise domain of 0.142c, where c is the hump chord length. Despite reasonable overall agreement, the predicted separation-bubble length was found to be larger thanthatintheexperiment. AmorerecentWRLESwasconductedforthesameproblembyYehetal.11 ThisWRLES was performed at half the experimental Reynolds number and for a spanwise domain of 0.2c, using 93 million grid points. Thepredictedseparation-bubblelengthwaslargerthantheexperimentalmeasurementbyabout20%. Well-resolved simulations of complex separated flows are very rare in the literature mainly because of the ex- tensive computational resources required. To address this deficiency and generate detailed data needed for a better understandingoftheproblemathand,WRLESoftheNASAhumpbenchmarktestcaseisperformedusingupto850 milliongridpointsinthepresentwork. Asystematicstudyregardingtheeffectofdomainspan,subgrid-scale(SGS) model,tunnelbackpressure,upstreamboundarylayerconditionsandgridrefinementontheresultsisperformed. As willbeseen,theseparation-bubblelengthisfoundtobesensitivetothespanofthecomputationaldomain. Thispaper 2of22 AmericanInstituteofAeronauticsandAstronautics willdiscussthemainproblemsthatwereencounteredduringthecourseofthisworkandpresentviablesolutions. II. ComputationalMethodology Thecodeusedinthepresentstudysolvestheunsteadyfullythree-dimensionalcompressibleNavier-Stokesequa- tionsdiscretizedonmulti-blockstructuredandoversetgrids. Aturbulencesimulationcanberunintheformofdirect numerical simulation (DNS), large-eddy simulation (LES), delayed detached eddy simulation (DDES) or unsteady Reynolds-AveragedNavier-Stokes(URANS)calculation. Thecodeemploysanoptimizedprefactoredfourth-ordercompactfinite-differencescheme12 tocomputeallspa- tial derivatives. This optimized scheme offers improved dispersion characteristics compared to standard sixth- and eighth-order compact schemes.13 To eliminate spurious high-frequency numerical oscillations that may arise from severalsources(suchasgridstretching,unresolvedfluctuationsandapproximationofphysicalboundaryconditions) and ensure numerical stability during the simulation, we also employ a sixth-order compact filtering scheme.14,15 Theoverset-gridcapabilityisusefulinmeshingcomplexgeometriesandavoidinggrid-pointsingularities. Tomain- tainhigh-orderaccuracythroughouttheentirecomputationaldomain,weperformasixth-orderaccurateexplicitLa- grangian interpolation16 whenever overset grids are used. A Beam-Warming type approximately factorized implicit schemeisusedforthetimeadvancement.17 Moredetailsofthesimulationmethodologyaregiveninrelatedpublica- tions.18–21 Successfulapplicationsofthemethodology tosomewall-bounded flowproblemscanbefoundinpapers byUzunandcoworkers.18,22,23 III. TestCase: FlowSeparationOverNASAWall-MountedHump Thetestcasestudiedinthispaperinvolveslow-speedflowseparationoverawall-mountedhumpgeometry,also knownastheNASAhump,representativeoftheuppersurfaceofanairfoil,asdepictedinFigure1(a). Inthefollow- ing sub-sections, we provide a description of the experimental setup and computational modeling, discuss the main problemsencounteredandexaminethesimulationresults. A. ExperimentalSetupandComputationalModeling AnexperimentalinvestigationoftheNASAwall-mountedhump,depictedinFigure1a,wasconductedbyGreenblatt etal.2 Thereferencelengthscaleistakenasthehumpchordlength, c = 420mm. TheReynolds number basedon freestreamvelocityandchordlengthhasavalueofRe ≈ 936,000. Aswillbediscussed,thereissomeuncertainty c regarding the momentum-thickness Reynolds number of the incoming boundary layer, Re , measured about two θ chordlengthsupstreamofthehump. TheexperimentalstudyoriginallyreportedavalueofRe =7200. Ourestimate θ for this parameter is Re ≈ 6454, as discussed in section III.B.1. Regardless of the exact value, such Re values θ θ are relatively high from a computational point of view, as the highest Re achieved by DNS24 and WRLES25 for θ incompressible turbulent boundary layers at the time of this writing is 6500 and 8300, respectively. Because of the highRe ,asignificantnumberofgridpointsisneededforthewall-resolvedcalculationsinthepresentstudy. θ ThefreestreamMachnumberupstreamofthehumpis0.1. Theexperimentalsetupincludesendplatesattached to the model as depicted in Figure 1a. Modeling of these end plates in a wall-resolved simulation would prove computationallyveryexpensive. Thesimulationsthereforeuseaperiodicspanwisedomain. Thesimulationsconsider twospansof0.2cand0.4c. Thedistancebetweentheendplatesintheexperimentis1.4c. Theflowblockageeffect causedbytheendplatesintheexperimentcausesadditionalflowaccelerationoverthehump. Aspeciallycontoured top wall that mimics this effect is therefore used here. Figure 2 depicts the main features of the problem under investigation and also shows the specially contoured top wall. More details can be found on the NASA Turbulence Modeling Resource websiteat: http://turbmodels.larc.nasa.gov/nasahump val.html a. The experimental chord-based Reynoldsnumberismatchedexactlyinallsimulations. Aswillbediscussed,theexperimentalMachnumberisalso matched exactly in the narrow-span case, but is increased to 0.2 for the wide-span case to speed up the calculations sincethecomputationalcostincreaseswithdecreasingMachnumberforourcompressibleflowsolver. Furtherdetails ofthesimulationsareprovidedinsectionIII.C. aWebsitelastaccessed25October2016. 3of22 AmericanInstituteofAeronauticsandAstronautics Figure2. MainfeaturesoftheNASAhumpproblemdepictedintermsofinstantaneousnormalizedstreamwisevelocitycontours. (Only partofthefullstreamwiseextentofthecomputationaldomainisshown.) B. ProblemsEncountered Beforeexaminingthecomputationalresults,wediscussanumberofimportantproblemsthatwereencounteredduring thecourseofthisworkandpresentpossiblesolutions. 1. ExperimentalIncomingBoundaryLayerDetails The first problem is concerned with the details of the incoming boundary layer measured at x/c = −2.14 in the experiment, where x denotes the streamwise distance measured from the hump leading edge (located at x/c = 0). Although the experimental study2 originally reported a momentum-thickness Reynolds number of Re = 7200 at θ thatlocation, themomentum-thicknessintegralofthemeanvelocityprofileobtainedfromaRANScalculation(that matchestheexperimentalmeanvelocityprofile26)providesalowervalueofRe ≈ 6454. Thecomparisonbetween θ thisRANSmeanvelocityprofileandtheexperimentalmeasurementisshowninFigure3. Moreover,theexperimental skin-frictionmeasurement27takenatx/c=−2.14providesanormalizedfrictionvelocityvalueofuτ/u∞ ≈0.0378. The corresponding Reynolds number based on the experimental boundary layer thickness and friction velocity is Re ≈ 2225. Foracanonicalflat-plateturbulentboundarylayer, thisfrictionvelocitycorrespondstoRe ≈ 5000. τ θ Flat-plate turbulent boundary layer data available from DNS24 and WRLES25 were examined to determine the Re θ corresponding to the reported skin-friction velocity. Figure 4 shows the flat-plate turbulent boundary layer profiles fromtheseDNSandLESfor5000≤Re ≤7000intheformofmeanstreamwisevelocityandstreamwiseturbulence θ intensity,andthecomparisonwiththeexperimentalmeasurement. Weseethatnoneofthecomputationalprofilesare anexactmatchtotheexperimental profile. Thesefindings suggestthattheboundary layerupstreamofthehumpin theexperiment(atx/c=−2.14)isperhapsnotpreciselyaflat-plateturbulentboundarylayer. Given this uncertainty in upstream boundary layer conditions, we considered two options for the simulations. ThefirstoptionistomatchthereportedupstreamskinfrictionintheexperimentandhencesetRe = 5000forthe θ incoming boundary layer upstreamof thehump. Aturbulent inflow generation technique thatgenerates acanonical flat-plate turbulent boundary layer at Re = 5000 can be employed for this purpose. The second option is to take θ themeanvelocityprofileavailablefromRANSandaddtheturbulentfluctuationstothismeanprofileusinganinflow generation technique. We use only the first option for the narrow-span calculation but explore both options for the wide-spancase. Fortheturbulentinflowgeneration,aspecificversionoftherescaling-recyclingtechnique,discussed inUzunandHussaini,28 isused. Thisinflowgenerationmethodisadditionallyaugmentedwithseveralmodifications proposedbyMorganetal.29 toeliminatepossibleenergeticlowfrequenciesthatmaybeartificiallyintroducedbythe turbulent inflow generation technique. The distance between the inlet and recycle stations is 15 times the incoming boundarylayerthickness. Thisdistanceisinthetypicallyrecommendedrange. 4of22 AmericanInstituteofAeronauticsandAstronautics 1 0.8 0.6 U / U wall hump experiment inflow ref RANS 0.4 0.2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 y / c Figure3. ComparisonbetweenRANS-predictedmeanstreamwisevelocityprofileandexperimentalmeasurementatx/c=−2.14. 2. GridSizeforWRLES Thesecondproblemisconcernedwithmaintainingareasonablegridresolutionwhilekeepingthetotalnumberofgrid pointsatamanageablelevelfortheavailablecomputationalresources. AproperWRLESrequiresasignificantgrid resolutioninthenear-wallregionoftheturbulentboundarylayer.Thisisbecauseenergy-containingsmalleddiesinthe near-wallregionneedtoberesolvedandarenotaccountedforaccuratelybytheSGSmodeling. Ourpreliminarytests onacanonicalflat-plateturbulentboundarylayershowthatthepresentmethodologyrequiresastreamwiseresolution of∆x+ ≈ 25,aspanwiseresolutionof∆z+ ≈ 12.5,andawall-normalresolutionof∆y+ ≈ 1onthewall,where thesuperscript+indicateswallunits,inordertopredictthewallskinfrictionaccurately. Thisresolutionapproaches thetypicalvaluesusedinDNSofturbulentboundarylayers.24 Aswillbeseen,thesevaluespredicttheskinfriction reasonablywell(withinafewpercent)intheattached-flowregionpriortoseparation. wall hump experiment inflow 30 wall hump experiment inflow 3 flat plate DNS, Re = 5000 flat plate DNS, Re = 5000 flat plate DNS, Re = 6500 flat plate DNS, Re = 6500 flat plate LES, Re = 7000 25 flat plate LES, Re = 7000 2.5 20 2 U+ u+ 15 rms1.5 10 1 5 0.5 0 10-1 100 101 102 103 104 0 10-1 100 101 102 103 104 y+ y+ (a) Meanstreamwisevelocityprofilesinwallunits. (b) Streamwiseturbulenceintensityprofilesinwallunits. Figure4. Comparisonofexperimentalinflowmeasurementswithflat-plateturbulentboundarylayerdatafromDNS24andLES25. Maintainingthisgridresolutionallthewaytotheedgeoftheturbulentboundarylayerinthepresentproblembe- comesprohibitivelyexpensivebecauseofcomputationalresourcelimitations.Areasonablecompromiseistomaintain thefinegridspacingsonlyinthenear-wallregion,sayuptoabouty+ ≈200,andthencoarsenboththestreamwiseand spanwiseresolutionbyafactoroftwointheouterregionawayfromthewall. Thisisthestrategyadoptedhere. Inthe near-wallregion,theturbulentboundarylayerapproachingthehumpisresolvedusing∆x+ ≈ 25,∆z+ ≈ 12.5and 5of22 AmericanInstituteofAeronauticsandAstronautics ∆y+ ≈ 0.8onthewall. Similarresolutionismaintainedfortheboundarylayeroverthehump. Theflowsolverhas overset-gridcapability,andtheinnerandouterregiongridscommunicatebymeansofsixth-orderaccurateoverset-grid interpolation. Figure5showsacross-sectionofthetwo-leveloverset-gridsystematastreamwiselocationupstream ofthehumpandtheinstantaneousboundarylayerstructuressuperposedontheoverset-grid. Thesmallstructuresnear the wall are resolved by the near-wall fine grid while the coarser outer grid seems appropriate for the larger struc- turesawayfromthewall. Asimilaroverset-gridstrategywassuccessfullytestedinthestudyofTollmien-Schlichting instabilitywavesdevelopinginasubsoniclaminarboundarylayer. Comparisonwiththesolutionoftheparabolized stabilityequationsshowedthattheoverset-gridtechniquedidnotintroduceanerrorinthecomputationofinstability waves. (a) Overset-gridsystem. (b) Boundary layer structures (depicted in terms of instantaneous streamwisevelocitycontours)superposedonoverset-grid. Figure5. Two-leveloverset-gridsystematx/c=−2.Forbetterclarity,theverticalscaleintheleftandrightfiguresisnotthesame. 3. AcousticResonanceandItsEffectonTurbulentInflowGeneration Thethirdproblemisconcernedwiththeturbulentinflowgenerationintheregionupstreamofthehump. Inourinitial simulations, we attempted to employ a rescaling-recycling technique28 to generate a turbulent incoming boundary layeratabouttwochordlengthsupstreamofthehump. However,thisstrategydidnotworkasexpectedinthisregion becauseitdidnotproduceaboundarylayerwithpropertiessimilartothoseofacanonicalflat-plateturbulentboundary layer. Uponfurtherinvestigation,anacousticresonancephenomenonwasdeterminedtoberesponsibleforthisissue. We found that the acoustic disturbances generated by the flow going over the hump get trapped inside the closed tunnel and excite an acoustic resonance, eventually giving rise to trapped waves in front of the hump, as depicted in Figure 6(a). These trapped acoustic waves continuously propagate up and down in front of the hump, precisely in the region where information is rescaled and recycled for the turbulent inflow generation. Upon impingement ontothelowerwall,theyinstantaneouslycreatelocaladverseorfavorablepressuregradients,thusviolatingthezero pressuregradientassumptiononwhichtheinflowgenerationtechniqueisbased. Wind-tunnelacousticresonanceisa phenomenonthatisknowntooccurinphysicalexperimentsandhaspreviouslymotivatedIkedaetal.30 toperforma computationalinvestigationofthewind-tunnelacousticresonanceinducedbytheflowovera2-Dairfoil. We considered two possible solutions to this problem. The first option is to perform an auxiliary simulation for aturbulentboundarylayerdevelopingonaflatplateandusetheunsteadyinformationatagivenstreamwisestation (where the local momentum-thickness of the boundary layer may correspond to a value such as Re = 5000) from θ this calculation as the inflow conditions for the hump simulation. This option was in fact employed in the narrow- span simulation. However, this strategy further increases the computational cost and complexity as it requires two simulationstobecarriedoutsimultaneously. Asecondandcomputationallycheaperoptionistosuppresstheacoustic resonance by adding a damping term to the right-hand side of the governing equations. The damping term is only activeinthevicinityofthetoptunnelwallandhastheformofσ(q−q),whereσisafunctionoftheverticaldistance andcontrolsthestrengthofthedampingterm.σiszerointheregionwherethedampingtermisinactive.Thedamping termforcesthenumericalsolution,q,inthechosenregiontowardareferencestate,suchasarunningtime-averageof thelocalflow,q. Theregioninwhichthedampingtermisappliedissufficientlyawayfromtheturbulence-containing region, hence the turbulent fluctuations in the attached and separated regions are not affected by this term. Figure 6(b)showsaschematicofthisapproachanddemonstratesthatthedampingtermisindeedeffectiveinsuppressingthe acousticresonance. Withthetroublesometrappedacousticwavesoutofthepicture,theinflowgenerationtechnique 6of22 AmericanInstituteofAeronauticsandAstronautics can work successfully in the region upstream of the hump and there is no need for an auxiliary simulation. This approachisusedinthewide-spancalculations. (a) Wavestrappedupstreamofhump. (b) Suppressionofacousticresonance. Figure6. Acousticresonancegivingrisetotrappedwavesanditssuppressionbyuseofaspecialdampingterm. C. SimulationDetails The experimental chord-based Reynolds number (i.e., Re = 936,000) is matched exactly in all simulations. The c simulationsconsidertwospansof0.2cand0.4c. TheexperimentalMachnumberof0.1isalsomatchedinthenarrow- spancalculation,butisincreasedto0.2forthewide-spancalculationstoreducethecomputationaltime.Thetimestep ofallsimulationsis2.5×10−4c/a ,wherea isthereferencespeedofsound. Thereferencefreestreamconditions ref ref aretakenatx/c = −2.14. Withthistimestep,themaximumCFLnumberisabout17. Threesubiterationspertime stepareappliedforthefullyimplicittimeadvancementscheme.OurexperienceshowsthatthemaximumCFLnumber inthesimulationshouldbekeptbelowamaximumofapproximately25toensuresufficienttemporalaccuracyfrom thesecond-orderimplicittimeadvancementscheme. Thiswasdeterminedbyexaminingtheskin-frictionpredictions inacanonicalflat-plateturbulentboundarylayerobtainedwithdifferentCFLnumbers. A schematic of the computational domain is shown in Figure 2. This figure depicts part of the full streamwise extentofthedomain. Toreiterate,thehumpleadingedgeislocatedatx/c=0,whiletheinletboundaryofthehump domainisatx/c = −2.14. Thephysicalregionofinterestinthehumpdomainextendsuptox/c = 1.6. Asponge zoneisplaceddownstreamofthephysicalregionofinterest. Theoutflowboundaryofthecomputationaldomainis placed at the end of the sponge zone and is located at x/c = 4. The back pressure on the outflow boundary is set slightlybelowtheupstreampressure(seesectionIII.D.2). Thetopwallofthecomputationaldomainistreatedasan inviscid wall while viscous adiabatic boundary conditions are imposed on the lower wall. Characteristic boundary conditions are applied at the outflow boundary. The inflow boundary is based on characteristic relaxation boundary conditions31 that inject turbulent fluctuations (generated by the inflow generation technique) in the boundary layer whileallowingupstream-travelingwavestoexitthedomain. Theturbulentboundarylayerapproachingthehumpis resolved using 90 to 100 points in the wall-normal direction. Flow acceleration over the hump causes a significant thinningoftheboundarylayer. Thethinnestpartoftheboundarylayeroverthehumpisresolvedusingaminimumof about30pointsinthewall-normaldirection. 1. Narrow-SpanCalculation Asmentionedearlier,thenarrow-spancalculationinvolvesanauxiliarysimulationthatcomputestheturbulentbound- ary layer developing over a flat plate and a main simulation for the flow over the hump. An instantaneous plane extracted from the flat-plate domain at Re = 5000 is injected as the inflow conditions on the upstream boundary θ of the hump domain. The flow solver runs on the two domains simultaneously and generates the inflow conditions forthehumpdomainonthefly. Theflat-platedomaincontainsabout90millionpointswhilethewallhumpdomain contains about 210 millionpoints. The two-level overset-grid strategy isused forboth domains. The staticVreman SGSmodel32isemployedwithafixedmodelcoefficientof0.025. Statisticalresultsareaveragedoverabout10chord flow times, where one chord flow time unit (i.e., c/U ) is defined as the time it takes for the reference freestream ref velocity,U ,totravelonechordlength. ref 7of22 AmericanInstituteofAeronauticsandAstronautics 2. Wide-SpanCalculations The wide-span grid is obtained by doubling the spanwise extent of the narrow-span grid while keeping the same resolutionasforthenarrow-spangrid. Inthewide-spancalculations,thetunnelacousticresonanceissuppressedby theuseofadampingterminthevicinityofthetopwallandtheturbulentinflowgenerationisdoneupstreamofthe humpwithinthemainsimulation. Thenumberofpointsinthewide-spangridisabout420million. Becauseofthe increased grid size, the Mach number is increased from 0.1 to 0.2 to speed up the calculation and help reduce the computationalcost. Thetimestepiskeptthesameasthatinthenarrow-spancase(2.5×10−4c/a ). Doublingthe ref Machnumberfrom0.1to0.2meansthattheflowinthewide-spancaseconvectstwiceasfast. Theinflowandoutflow boundary locations are the same as those in the narrow-span case. A number of calculations are performed for the wide-span case. These calculations reveal the effect of SGS model, tunnel back pressure, upstream boundary layer Re andgridrefinementonthenumericalpredictions. Statisticalresultsaretypicallyaveragedoverabout30c/U . θ ref D. Results Wenowexaminethesimulationresultsandmakecomparisonswiththeexperimentalmeasurements. 1. Narrow-SpanCalculation The results from the narrow-span WRLES are analyzed first. Figure 7 depicts the complex nature of the separated flow in the aft portion of the wall-mounted hump. The initially thin separated free shear layer quickly gives rise to the formation of large-scale structures, which in turn govern the dynamics of the shear layer growth and dictate the reattachment location of the separated flow. Figure 8 shows the skin-friction and pressure coefficients and the comparisonwiththeexperimentalmeasurement. Theskin-frictionandpressurecoefficientsaredefinedas τ p−p C = wall and C = ref (1) f 1ρ U2 p 1ρ U2 2 ref ref 2 ref ref where τ is the viscous wall shear stress, ρ and p, respectively, are the density and pressure, and the subscript wall ref denotes the reference freestream conditions at x/c = −2.14. The WRLES results are based on the mean flow obtained by averaging the unsteady flow in time and along the span. The skin-friction comparison figure includes the error bars for the experimental data. The overall C distribution and the separation-bubble length are predicted f by the simulation reasonably well. The separation and reattachment locations in the simulation are determined by the streamwise locations at which C becomes zero. The predicted separation and reattachment locations based on f thiscriterionarefairlyclosetotheexperimentallyobservedvalues. Theseparationlocationisatx/c ≈ 0.665inthe experiment andatx/c ≈ 0.659 inthesimulation, whilethereattachment isatx/c ≈ 1.11 intheexperiment andat x/c = 1.095inthesimulation. TheunderpredictionofC inthepeakregionpriortoflowseparationisprimarilyan f SGSmodeleffect,aswillbedemonstratedinthewide-spancalculations. Of particular note is the plateau in the measured skin friction at 0.1 < x/c < 0.2, which is predicted well by the present simulation. This plateau is presumably caused by a tendency toward relaminarization due to the strong favorablepressuregradientinthisregion. Figure8aincludestherelaminarizationparameter,K,andshowsthatthis parameter reaches its peak just before the plateau in C . The original definition of the relaminarization parameter f is K = ν/U2 (∂U /∂s), where ν is the kinematic viscosity, U is the boundary layer edge velocity and s is the e e e surface d¡istance¢. For an incompressible flow, the following equation was derived for K in terms of Cp and the ReynoldsnumberbyBourassa34 usingBernoulli’sequationandboundarylayerassumptions: 3/2 1 1 1 ∂C K =− p where s∗ =s/c (2) 2Re ·1−C ¸ ∂s∗ c p TherelaminarizationparametershowninFigure8aiscomputedusingthismoreconvenientexpression. Thegenerally acceptedcriticalvalueofK abovewhichrelaminarizationcantakeplace35 isabout3×10−6. ThepeakvalueofK inFigure8a(K ≈ 4.87×10−6)isgreaterthanthiscriticalvalue. PreviousRANSandWMLEScomputationshave missedthisplateauinC becauseoftheirinherentinabilitytocapturerelaminarization. f Figure8bincludestheexperimentalC measurementstakenwithandwithouttheendplates,andshowsthecom- p parisonwiththecomputationalresults. Sincethesimulationincludesaspeciallycontouredtopwallthatapproximates theend-plateeffects,thecomparisonagainsttheexperimentperformedwiththeendplatesismoreappropriate. The C comparisondisplaysreasonableagreementbetweenthesimulationandtheexperiment. Althoughthesimulation p 8of22 AmericanInstituteofAeronauticsandAstronautics Figure7. Complexstructureofseparatedflowintheaftportionofthewall-mountedhumpshowninFigure1. Instantaneousiso-surface ofQ-criterion33(constantQ=15aref/c)coloredbystreamwisevelocityisplotted. accurately captures the primary suction peak caused by flow acceleration over the hump, the secondary peak region withintheseparationbubbleissomewhatunderpredictedbythesimulation. Aswillbeseen,thisunderpredictionof the secondary peak is also present in the wide-span calculations. This suggests that the end plates attached to the experimentalmodelmayhavesomeeffectontheseparationregion,whichcannotbecapturedinaspanwise-periodic simulation with the specially contoured top wall. This is supported by the recent findings of Duda and Fares6 who performedawall-modeledsimulationoftheproblemincludingtheendplatesandobtainedbetterC comparisonwith p theexperiment,althoughsomedifferencesremainedinthesecondarypeak. Thatsimulationcapturedcornervortices atthejunctionoftheendplatesandthehump,whichwerefoundtoaffecttheshapeandsizeoftheseparatedregion. Despiteanaccuraterepresentationoftheexperimentalsetupintheirsimulation,theC curvemissedtherelaminariza- f tionplateau,overpredictedthepeakregionpriortoseparationandneededfurtherimprovementbothintheseparated andattachedregions. Figures9and10,respectively,plotthemeanvelocityandReynoldsstresscomparisonsatanumberofstreamwise stations. The WRLES profiles are obtained by averaging the unsteady flow in time (over 10 chord flow times) and alongthespan. Thefirststationislocatedatx/c = 0.65,whichisjustupstreamoftheseparationlocation. Thelast stationislocatedatx/c = 1.3, whichisdownstreamofthereattachmentlocationatx/c ≈ 1.11. Theexperimental data shown in the comparisons have been obtained using two-dimensional particle image velocimetry (PIV) on the centralplane. Forclarity,ahorizontalshiftisappliedtotheprofilesdisplayedineachsubfigure. Therespectiveshift noted for each subfigure denotes the distance between the major ticks on the horizontal axis. These comparisons displaygoodoverallagreementbetweenthesimulationandexperiment. TheReynoldsstressprofilesinthesimulation appeartobemoreenergeticthantheexperimentatthefirstcomparisonstation. Wenotethattheattachedboundary layerjustupstreamofseparationisratherthinandtheexperimentalPIVmeasurementisnotsufficientlyaccurateinthe near-wallregion. TheseprofilesshowthatflowseparationisaccompaniedbyarapidgrowthofReynoldsstressesand thickeningoftheseparatedshearlayer. Theseparatedshearlayerexpandswithstreamwisedistanceandsubsequently reattachesonthelowerwall. Althoughtheseinitialresultsfromthenarrow-spanWRLESareveryencouraging,thecalculationsperformedon a domain with a doubled span predict an earlier reattachment of the separated flow, as will be seen. We discuss the findingsfromthewide-spancalculationsnext. Theobservationsmadefromthewide-spanWRLESsuggestthatthe seeminglygoodpredictionsobtainedinthenarrow-spanWRLESmighthavebeenfortuitous. 9of22 AmericanInstituteofAeronauticsandAstronautics experiment (with end plates) 0.008 narrow-span WRLES 6E-06 relaminarization parameter, K 0.006 4E-06 0.004 Cf 2E-06K 0.002 0 0 -0.002 -2E-06 -0.5 0 0.5 1 1.5 x / c (a) Cf distribution. experiment with end plates experiment without end plates -1 narrow-span WRLES -0.8 -0.6 -0.4 p C -0.2 0 0.2 0.4 -0.5 0 0.5 1 1.5 2 x / c (b) Cpdistribution. Figure8. Skin-frictioncoefficient(Cf)andpressurecoefficient(Cp)comparisonsforthenarrow-spanWRLES. 10of22 AmericanInstituteofAeronauticsandAstronautics