SIMULATION OF SYNTHETIC JETS IN QUIESCENT AIR USING UNSTEADY REYNOLDS AVERAGED NAVIER-STOKES EQUATIONS VeerN.Vatsa∗ NASALangleyResearchCenter,Hampton,VA EliTurkel† Tel-AvivUniversity,IsraelandNIA,Hampton,VA 1 Abstract WereportresearchexperienceinapplyinganUnsteadyReynolds-AveragedNavier-Stokes(URANS)solverfortheprediction oftime-dependentflowsinthepresenceofanactiveflowcontroldevice. Theconfigurationunderconsiderationisasynthetic jet created by a single diaphragm piezoelectric actuator in quiescent air. Time-averaged and instantaneous data for this case wereobtainedatLangleyResearchCenter,usingmultiplemeasurementtechniques. Computationalresultsforthiscaseusing one-equationSpalart-Allmarasandtwo-equationMenter’sturbulencemodelsarepresentedherealongwithcomparisonswith theexperimentaldata. Theeffectofgridrefinement,preconditioningandtime-stepvariationarealsoexamined. 2 Introduction Significantinteresthasbeengrowingintheaerospacecommunityinthefieldofflowcontrolinrecentyears. AnentireAIAA conferenceisnowdevotedeveryotheryeartothisfield.Inparticular,aninternationalworkshop,CFDVAL2004[1],washeldin March2004atNASALangleyResearchCentertoassessthestate-of-the-artformeasuringandcomputingaerodynamicflows in the presence of synthetic jets. Thomas et al. [2] have conducted an exhaustive and comprehensive survey identifying the feasibilityofusingactiveflowcontroltoimprovetheperformanceofbothexternalandinternalflows. Suggestedapplications cover a wide range from smart materials and micro-electro-mechanical systems (MEMS) to synthetic zero net mass jets for enhancingcontrolforces,reducingdrag,increasingliftandenhancingmixingtonameafew. Itisalsoconjecturedthatactive flowcontrolwouldpermittheuseofthickerwingsectionsinnon-conventionalconfigurations,suchastheBlendedWingBody (BWB)configurationwithoutcompromisingtheaerodynamicperformance. Mostoftheresearchintheareaofactiveflowcontrolisofempiricalnature,mainlyduetothecostandlackofconfidence incomputationalmethodsforsuchcomplexflows. However,withouttheavailabilityofefficientandwell-calibratedcomputa- tionaltools, itwillbeaverydifficult, expensiveandslowprocesstodeterminetheoptimumlayoutandplacementforactive flowcontroldevicesinpracticalapplications. Withthecontinuousreductionofcomputercostsinrecentyears,moreattention isbeingdevotedtothesimulationofsuchunsteadyflows,andmanyresearchershaverecentlystartedexaminingvariousflow controldevicesfromacomputationalpointofview(Refs. [3]-[8]). Withfewexceptions, mostofthenumericalstudiesare undertakenwithoutanactiveinteractionwithexperimentalinvestigators. Comparisonswithexperimentaldataaresometimes doneyearsaftertheexperimentaldatahavebeenacquired. Undersuchascenario, onehastoreconstructsomeofthedetails abouttheexperimentalarrangementandboundaryconditionswithoutthebenefitofconcreteandconsistentinformation. Based onourexperiencefrompreviousvalidationexercises[9],werecognizedtheneedforactivecollaborationofthecomputational andexperimentalresearch. Withoutasymbioticrelationshipamongsuchgroups,majormisunderstandingscandevelopwhen ∗SeniorMember †Professor,DepartmentofMathematics,AssociateFellow 1 resultsfromthesedisciplinesindicatesignificantdifferences. Wewereveryfortunatetohaveafrankandcooperativerelation- shipwiththeresearchersconductingtheexperimentsaswellasaccesstopertinentexperimentaldata. Our primary objective for this work is to calibrate an existing computational scheme with experimental data for time- dependent flows encountered in active flow control environments. Special attention is devoted to establish appropriate and stableboundaryconditionsforsuchflows,especiallyintheabsenceofdetailedexperimentaldatarequiredforclosure. The configuration chosen for CFD validation is identified as Case 1 in the CFDVAL2004 workshop [1], and represents an isolated synthetic jet formed by a single diaphragm piezoelectric actuator exhausting into ambient quiescent air. Multiple measurement techniques including PIV, LDV and hot-wire probes were used to generate a large body of experimental data for this configuration. The details of the experimental setup and geometric configuration are described in Refs. [1, 10]. In thispaper,weassesstheeffectsofgridrefinement,preconditioningandturbulencemodelsontheflowfieldgeneratedbythis syntheticjetflowcontroldevice. Wereplacetheactuatorcavitywithasimplerconfiguration. Wedemonstrateandcalibrateour computationalmethodforsimulatingsyntheticjetsbycomparingthenumericalresultswiththeexperimentaldata. 3 Governing Equations A generalized form of the thin-layer Navier-Stokes equations is used to model the flow. The equation set is obtained from thecompleteNavier-Stokesequationsbyretainingtheviscousdiffusiontermsnormaltothesolidsurfacesineverycoordinate direction. Forabody-fittedcoordinatesystem(ξ,η,ζ)fixedintime,theseequationscanbewrittenintheconservativeformas: ∂(U) ∂(F−F ) ∂(G−G ) ∂(H−H ) Vol + v + v + v =0 (1) ∂t ∂ξ ∂η ∂ζ whereUrepresentstheconservedvariablevector.ThevectorsF,G,H,andF ,G ,H representtheconvectiveanddiffusive v v v fluxesinthethreetransformedcoordinatedirections,respectively.InEqn.(1),Volrepresentsthecell-volumeortheJacobianof thecoordinatetransformation. Amultigrid-based,generalpurposemulti-blockstructuredgridapproachisusedforthesolution of the governing equations. In particular, the TLNS3D flow code is used in this study to solve Eqn. (1). Several references existdescribingthediscretizationandalgorithmicdetailsofTLNS3Dcode. Weincludeonlyabriefsummaryofthegeneral features,andrefertotheworkofVatsaandco-workers[11,12]forfurtherdetailsregardingtheTLNS3Dcode. 4 Spatial Discretization ThespatialtermsinEqn. (1)arediscretizedusingacell-centeredfinitevolumescheme. Theconvectiontermsarediscretized using second-order central differences with matrix artificial dissipation (second- and fourth- difference dissipation) added to suppresstheodd-evendecouplingandoscillationsinthevicinityofshockwavesandstagnationpoints[13,14,15].Theviscous termsarediscretizedwithsecond-orderaccuratecentraldifferenceformulas[11]. Thezero-equationmodelofBaldwin-Lomax [16],one-equationmodelofSpalart-Allmaras[17]andMenter’stwo-equationSSTmodel[18]areavailableinTLNS3Dcode forsimulatingturbulentflows. Forthepresentcomputations,theSpalart-Allmaras(SA)modelandtheMenter’sSSTmodelare usedforsimulatingturbulentflows. 5 Temporal Discretization Regroupingthetermsonrighthandsideintoconvectiveanddiffusiveterms,Eqn. (1)canberewrittenas: dU =−C(U)+D (U)+D (U) (2) dt p a where C(U), D (U), and D (U) are the convection, physical diffusion, and artificial diffusion terms, respectively. The p a cell-volumeortheJacobianofthecoordinatetransformationisincludedintheseterms. Thetime-derivativetermcanbeapproximatedtoanydesiredorderofaccuracybyaTaylorseries dU 1 = [a Un+1+a Un+a Un−1+a Un−2+...] (3) dt ∆t 0 1 2 3 2 The superscript n represents the last time level at which the solution is known, and n+1 refers to the next time level to whichthesolutionwillbeadvanced. Similarly,n-1referstothesolutionatonetimelevelbeforethecurrentsolution. Eqn. (3) representsageneralizedbackwarddifferencescheme(BDF)intime,wheretheorderofaccuracyisdeterminedbythechoiceof coefficientsa ,a ,a ... etc. Forexample,a =1.5,a =−2anda =.5,resultsinasecondorderaccuratescheme(BDF2) 0 1 2 0 1 2 intime,whichistheprimaryschemechosenforthisworkduetoitsrobustnessandstabilityproperties[19]. Regroupingthe time-dependenttermsandtheoriginalsteady-stateoperatorleadstotheequation: a E(Un,n−1,..) 0Un+1+ =S(Un) (4) ∆t ∆t whereE(Un,n−1,..)dependsonlyonthesolutionvectorattimelevelsnandearlier. Srepresentsthesteadystateoperatoror therighthandsideofEqn.(2). Byaddingapseudo-timeterm,wecanrewritetheaboveequationas: ∂U a E(Un,n−1,..) + 0Un+1+ =S(Un) (5) ∂τ ∆t ∆t 6 Solution Algorithm Thealgorithmforsolvingunsteadyflowreliesonthesteady-statealgorithmintheTLNS3Dcode[11,12]. Thebasicalgorithm consists of a five-stage Runge-Kutta time-stepping scheme for advancing the solution in pseudo-time, until the solution con- vergestoasteadystate. Efficiencyofthisalgorithmisenhancedthroughtheuseoflocaltime-stepping,residualsmoothingand multigridtechniquesdevelopedforsolvingsteady-stateequations. BecausetheMachnumberinmuchofthedomainisvery lowweconsidertheuseofpreconditioningmethods[26,27]. In order to solve the time-dependent Navier-Stokes equations (Eqn. 5), we add another iteration loop in physical time outside the pseudo-time iteration loop in TLNS3D. For fixed values of S(Un), E(Un,n−1,..), we iterate on Un+1 using the standardmultigridprocedureofTLNS3Ddevelopedforsteady-state, untilthedesiredlevelofconvergenceisachieved. This strategy,originallyproposedbyJameson[20]forEulerequationsandadaptedfortheTLNS3DviscousflowsolverbyMelson et. al [19], is popularly known as the dual time-stepping scheme for solving unsteady flows. The process is repeated until thedesirednumberoftime-stepshavebeencompleted. ThedetailsoftheTLNS3Dflowcodeforsolvingunsteadyflowsare availableinRefs. [19,25,28]. 7 Boundary Conditions The boundary conditions required for solving the Navier-Stokes equations, such as the no-slip, no injection, fixed wall tem- peratureoradiabaticwall,far-fieldandextrapolationconditionsarewellunderstoodandreadilyavailableinmostflowcodes includingtheTLNS3Dcode. Themostaccurateproceduretosimulatethemovingdiaphragmwouldrequiremovinggridca- pability. For simplicity, we chose to simulate this type of boundary condition by a periodic velocity transpiration condition. Thefrequencyofthetranspirationvelocityatthediaphragmsurfaceinthenumericalsimulationcorrespondstothefrequency oftheoscillatingdiaphragm. Thepeakvelocityatthediaphragmsurfacewasobtainedfromnumericaliterationtomatchthe experimentallymeasuredpeakvelocityofthesyntheticjetemanatingfromtheslotexit. Thepressureatthemovingdiaphragm isalsorequiredforclosure.However,intheabsenceofunsteadypressuredatafromtheexperiment,weimposedazeropressure gradientatthediaphragmboundary. Wealsotestedthepressuregradientboundaryconditionobtainedfromone-dimensional normal momentum equation [21], which had very little impact on the solutions. Due to the simplicity and robustness, we selectedthezeropressuregradientboundaryconditionatthediaphragmsurface. 8 Synthetic Jet Test Case: Background Thetestconfigurationexaminedinthispapercorrespondstoasinglediaphragmpiezoelectricactuatoroperatinginquiescentair. Theoscillatorymotionofthediaphragmproducesasyntheticjetthatexhaustsintosurroundingair. Thisconfiguration,shown in Fig. 1, consists of a 1.2 mm wide rectangular slot connected to a cavity with a piezoelectric diaphragm, and corresponds to case 1 of the CFDVAL2004 workshop on flow control devices [1]. The cavity and diaphragm geometry of this actuator are highly three-dimensional in the interior. However, the actual slot through which the fluid emerges is a high aspect ratio 3 rectangularslotandcanbemodeledasatwo-dimensionalconfiguration. Partialviewofthe2-Dgridprovidedtotheworkshop participants, and used by the present authors is shown in Figs. 2(a) and 2(b). The computational results contributed by the workshop participants are available in refs. [1, 22]. A consensus developed during the workshop that simulating the flow field inside the actuator cavity with an oscillating piezoelectric diaphragm from first principles was beyond the capability of the existing CFD codes. Most of the workshop participants, including the present authors modeled the internal cavity of the actuator as a two-dimensional configuration, and simulated the diaphragm motion via a transpiration condition imposed at thediaphragmsurface. Someoftheworkshopparticipantsfurthersimplifiedthecavitymodelingbyimposingatranspiration condition at the bottom part of the slot’s neck or even directly at the slot exit. After examining these results, we concluded that as long as the unsteady velocity signal at the slot exit replicates experimental conditions, details of the cavity modeling haveaninsignificanteffectonthedevelopmentofthesyntheticjetemanatingfromtheslot. Anotherconclusionderivedfrom theCFDVAL2004workshopwasthatnoparticularmethodologyorturbulencemodelemergedsuperiorforsimulatingthistest case[1,22]. One of the major difficulties identified during the CFDVAL2004 workshop was the large disparity in experimental data obtainedfromPIV,hot-wireprobesandLDVmeasurementtechniques. Suchavariationinexperimentaldatamadeitdifficult tovalidatethenumericalmethods.Partofthedifficultyinacquiringaconsistentsetofexperimentaldataarosefromthefactthat theperformanceofthepiezoelectricdiaphragmdependsonambientconditions. Therefore,itsperformancedegradesovertime whichmeansthatforagiveninputvoltage,theactuatorproducessmallerjetvelocitiesasitages. Becausetheseexperiments wereconductedoveraperiodofseveralmonths,inconsistenciescreepedinthedata. We include here sample results obtained with the TLNS3D code to encapsulate the status of CFD simulations for this configurationattheconclusionoftheCFDVAL2004workshop.Forthesecomputations,weusedthe2-Dgridofapproximately 65,000nodesfromtheCFDVAL2004workshopwebsiteasthebaselinegrid. AsdepictedinFigs. 2(a),2(b),thisgridincludes the internal cavity and the diaphragm geometry. The transpiration condition is imposed at the diaphragm surface, which is represented by the vertical line on the left side at the bottom of Fig. 2(b). We present the time-history of the (vertical) v- velocityatx=0,y =0.12mm,andthetimeaverageofthev-velocityalongthejetcenterlineinfigures3and4,respectively. Thecomputationswereperformedwithseveralgriddensities,twophysicaltimestepsandtwoturbulencemodels. Thecoarse grid(cg)wasobtainedbyeliminatingeveryotherpointfromthebaselinegrid,andafinegrid(fg)wasobtainedbyadding50% pointsinthenormaldirection. Weseefromthesefiguresthatvaryingallthesefactorsdidnotproduceanymajorchangesin thecomputationalresultsinregionneartheslotexit. However,furtherawayfromtheslotexit,coarsegrid(cg)resultsindicate thatthecoarsergriddoesnotprovideadequateresolutionforthisproblem. Afinergrid(fg)andareductionintimestep(low dt)hadaninsignificanteffectonthecomputationalresults. BecauseofthescatterbetweentheexperimentalPIVandhotwire data,itisdifficulttomakeanydefinitiveconclusionregardingtheaccuracyofturbulencemodels. 9 Results Yaoet.al[10]haverecentlyrevisitedthesyntheticjettestcaseandacquiredexperimentaldataforthisconfigurationwithanew diaphragm. The detailed field data was obtained with the PIV technique. In addition, pointwise data along the jet centerline wasobtainedwithhotwireandLDVtechniques. Theperformanceoftheactuatorwasmonitoredregularly. Theresultsfrom thisstudyshowgoodconsistencyamongmultiplemeasurementtechniques[10]. Wesimulatedthenewexperimentaltestcasewithasimplifiedcavitygeometry,shownschematicallyinFig. 5. Thetran- spirationconditionisimposedatthebottomoftheslot’snecktosimulatethevelocitygeneratedbytheoscillatingdiaphragm. SimilarboundaryconditiontreatmentproducedsatisfactoryresultsattheCFDVAL2004workshop,andhasalsobeenstudied indetailbyYamaleevandCarpenter[23]. Theydemonstratedthatforactuatorswithdeepcavities,specifyingthetranspiration condition at a distance of at least 4-5 slot widths away from the slot exit produces only a small loss in numerical accuracy. Atop-hatvelocityprofile,withadominantfrequencyof450Hz.,replicatingtheexperimentalconditionswasimposedatthe bottomboundary. Thepreciseformofthevelocitysignalwasobtainedbycurvefittingthemeasuredvelocitiesattheslotexit (x=0, y=0.3 mm) with a Fast Fourier Transform to reflect the proper mode shape and to ensure zero net mass transfer. The amplitude of this velocity was determined numerically to match the peak velocity from the experiment at the slot exit. The freestreamMachnumberintheexteriorquiescentregionisspecifiedasM =.001forsimulatingincompressibleflowinthe ∞ compressibleflowcodetoavoidnumericaldifficultiesatMachzero. Thecomputationalgridforthiscasewasobtainedfromthebaselinegriddescribedintheprevioussectionbyeliminating theportionsofgridbelowtheneckoftheslot. Theresultinggridconsistsofover60,000nodes,andshouldprovideadequate resolutionbasedonthegridrefinementstudyreportedinreference[1],andasseenfromtheresultsinfigures3and4.Similarly, 4 72time-steps/periodcorrespondingto5◦ phaseanglebetweenthetimestepsshouldprovideadequatetemporalresolutionfor thisproblem. Theone-equationSpalart-Allmaras(SA)[17]turbulencemodelwasusedtosimulatetheeffectofturbulencein thesecomputations. Basedonthepeakjetvelocityandslotwidth,theReynoldsnumberisapproximately3000,andtherefore fallsintheregimewherethejetisexpectedtobeturbulent. Therefore,theflowwasassumedtobefullyturbulentinthepresent simulations. Thetime-historyoftheverticalvelocityforacompleteperiodfromthecomputationalresultswithSAmodeliscompared with the experimental data in Fig. 6 at x=0 and y=0.3 mm. This is the closest point to the slot exit where the PIV data is available. In addition to the PIV data, LDV measurements are also available at this location and are shown here. The LDV data was scaled down by a factor of 0.9 as suggested by Yao et. al [10] to match the diaphragm displacement for the two setsofmeasurements,andisingoodagreementwiththePIVdataexceptforaflatterregionnearthe150◦ phase. Theoverall agreementbetweenthecomputationalandexperimentalresultsisquitegoodatthislocation,andthereforeservesasanaccurate boundarydataforthejetemergingfromtheslot. Next, we compare the time-averaged v-velocities along the jet centerline in Fig. 7 with the PIV and LDV data. The experimental data from two different techniques (PIV and LDV) is found to be in good agreement with each other, giving credibilitytotheaccuracyandconsistencyofthemeasurementsforthiscase. TheoverallagreementofthebaselineTLNS3D resultswiththeexperimentaldataisalsoquitegood.Thelow-speedpreconditioningresultsincludedinthisfigureareessentially identical to the baseline results. Because low-speed preconditioning ([26]-[28]) primarily reduces the artificial viscosity for unsteady flows, it is inferred that the artificial viscosity is already low in these simulations. The time-averaged v-velocity profilesaty=1and4mmareshowninFig. 8andFig. 9,respectively. Exceptforasmallervelocitypeakatthecenterlineat y=1mm,thecomputationalresultsareinverygoodagreementwiththeexperimentaldata. Thecontourplotsofthetime-averagedv-velocitiesbasedonPIVmeasurementscoveringadistanceof8mmfromslotexit areshowninFig. 10(a). NotethatthehighresolutionPIVdatawaslimiteduptoadistanceof8mmfromslotexitbecause ofcostandtimeconstraints. ThecontourplotsfromourbaselinecomputationsareshowninFig. 10(b)forcomparison. The computationalresultsappeartoaccuratelycapturealloftheprominentfeaturesseeninthePIVdataincludingthewidthand spreadingrateofthesyntheticjet. Wenowexaminethephase-averagedvelocitiesatselectedlocationsinspaceandtime,startingwithv-velocitiesaty=2and 4mmalongthejetcenter-line. ThePIVandLDVdataalongwithbaselineTLNS3Dsolutionsareshownattheselocationsin Figs. 11(a)and11(b). Thecomputationalresultsareinfairlygoodagreementwiththetwosetsofexperimentaldata,especially in the suction phase. The agreement with the experimental data further away from the slot exit is slightly worse during the peakexpulsioncycle. Inparticular,theCFDresultspredictadelayedphaseshiftforthepeakexpulsion,reflectiveofsmaller convectivespeedforoutwardmovementofthevortexcomparedtotheexperimentaldata. Wegainabroaderperspectiveoftheflow-fieldbyexaminingthecontourplotsofthevelocitiesatthephaseanglesrepresen- tativeoftheexpulsion(phase=75◦)andsuction(phase=255◦)cycles. Thevelocitycontoursforstreamwise(u-vel)andvertical velocities(v-vel)obtainedfromthePIVdataandTLNS3DcomputationalresultsareshowninFigs. 12-15andFigs. 16-19. ThesefiguresweregeneratedusingidenticalcontourlevelsforboththeexperimentalandCFDdataforprovidingaquantitative comparisons. Notethatsolidlinesrepresentpositivevalues,whiledashedlinerepresentnegativevaluesforthevelocities. This signconventionishelpfulinidentifyingtheflowdirectionandthepositionofthevortexcenter. Itisclearfromthesefigures thatthecomputationalresultscapturemostofthepertinentfeaturesobservedexperimentallyandareinverygoodagreement for the suction phase. During the expulsion phase, the computed vortex center is located closer to the slot exit compared to experimentaldata,althoughthepeakvelocityatthevortexcenterisingoodagreementwiththePIVdata. Yaoetal. [10]have observedincreasingthree-dimensionaleffectsforthiscaseasonemovesawayfromtheslotexit, mainlyduetoringvortices formedfromtheslotedges. Weconjecturethattheseringvorticesinduceforcesthatacceleratetheconvectionofsyntheticjet inthefarfield.Wheteherthisindeedistheprimaryreasonfordifferencesinconvectivespeedofthevortex,canonlybeverified by3-Dsimulationswithsufficientresolutiontocapturethevorticalstructureemanatingfromtheedgesoftheslot. 10 Concluding Remarks Detailed comparisons have been presented for time-averaged and phase-averaged velocities between experimental data and CFDresults. Theeffectoftruncationerrorswerefoundtobesmallbasedonagridrefinement,preconditioningandphysical time-steprefinementstudies.Thedifferencesbetweenthesolutionsobtainedfromtheone-equationturbulencemodelofSpalart andtwo-equationmodelofMenterwerefoundtobesmallinthenearfieldforthesyntheticjet. Themodelingoftheinternal flow in thecavity of the actuator turned out be anextremely difficult problem. Fortunately, thedevelopment of the synthetic 5 jetinthequiescentmediumisdrivenprimarilybythevelocityfieldattheslotexit,anddetailedmodelingofthecavityisnot warranted. Based on comparisons of the computational results with the original experimental data, it was decided to repeat the experiments to obtain more consistent data. The computational results in the reduced domain with a modified forcing functionarefoundtobeinmuchbetteragreementwiththenewexperimentaldatainthenearfield. However, theagreement withtheexperimentaldatadeterioratesinregionsfurtherawayfromtheslotexit. Basedontheavailableexperimentaldata,it appears that the flow becomes three-dimensional after 5-6 slot widths away from the exit. 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[28] Vatsa,V.N.andTurkel,E.:“AssessmentofLocalPreconditionersforsteadystateandtimedependentflows”.AIAApaper 2004-2134,June2004. 7 Figure1: SchematicofPiezoelectricActuator 12 11 0 10 9 -10 8 7 -20 m) m) m6 m ( (-30 Y5 Y 4 -40 3 2 -50 1 0 -60 -4 -3 -2 -1 0 1 2 3 4 -20 -15 -10 -5 0 5 10 15 20 X(mm) X(mm) (a)Globalview (b)Detailedview Figure2: ComputationalgridforPiezoelectricActuator 8 Exp.data,PIV 40 Exp.data,Hotwire 10 NASA-tlns3d-sa NASA-tlns3d-sa(cg) NASA-tlns3d-sa(fg) NASA-tlns3d-sa(lowdt) 8 20 NASA-tlns3d-sst 6 s s m/ 0 m/ v, v, 4 Exp.data,PIV Exp.data,Hotwire NASA-tlns3d-sa -20 NASA-tlns3d-sa(cg) 2 NASA-tlns3d-sa(fg) NASA-tlns3d-sa(lowdt) NASA-tlns3d-sst -40 0 0 90 180 270 360 0 5 10 15 20 phase,deg y,mm Figure 3: Time-history of v-velocity at x=0, y=0.12 Figure 4: Time-averaged v-velocity along jet center- mm line 40 20 0 Y -20 -40 -60 -40 -20 0 20 40 X Figure5: Simplifiedmodelofactuator 9 10 40 PIV data LDV data TLNS3D 8 30 c) e s 20 m/ V−vel (m/sec) 100 nterline velocity ( 46 Exp. data, PIV −10 ce Exp. Data, LDV TLNS3D−sa (no prec) 2 TLNS3D−sa (with low−speed prec) −20 −30 0 0 90 180 270 360 0 4 8 12 16 20 Phase angle (deg) y (mm) Figure6:Time-historyofv-velnearslotexit,SynjetII Figure7: Averagev-velalongcenterline,SynjetII 12 12 PIV data PIV data 10 TLNS3D (no prec) 10 TLNS3D (no prec) TLNS3D (low−speed prec) TLNS3D (low−speed prec) 8 8 6 6 4 4 2 2 0 0 −2 −2 −4 −4 −6 −2 2 6 −6 −2 2 6 Figure8: Averagev-velaty=1mm,SynjetII Figure9: Averagev-velaty=4mm,SynjetII 10