Andrew P. S. Wheeler1e-mail: a.wheeler@qmul.ac.ukSchool of Engineering and Materials Science,Queen Mary, University of London,Mile End Road,London, E1 4NS, UKNicholas R. AtkinsWhittle Laboratory,University of Cambridge,Cambridge, CB3 0DY, UKLi HeDepartment of Engineering Science,University of Oxford,Parks Road, Oxford OX1 3PJ, UKTurbine Blade Tip Heat Transfer inLow Speed and High Speed FlowsInthispaper,highandlowspeedtipowsareinvestigatedforahigh-pressureturbineblade. Previousexperimentaldataareusedtovalidateacomputationaluiddynamics(CFD) code, whichisthenusedtostudythetipheat transferinhighandlowspeedcascades.TheresultsshowthatatenginerepresentativeMachnumbers,thetipowispredominantly transonic. Thus, compared with the low speed tip ow, the heat transfer isaffected by reductions in both the heat-transfer coefcient and the recovery temperature.ThehighMachnumbersinthetipregionM1.5 leadtolargelocal variationsinrecoverytemperature. Signicant changesintheheat-transfercoefcient arealsoob-served.Theseareduetochangesinthestructureofthetipowathighspeed. Athighspeeds, thepressuresidecornerseparationbubblereattachmentoccursthroughsuper-sonicacceleration, whichhalvesthelengthof thebubblewhenthetip-gapexit Machnumber is increasedfrom0.1to1.0. Inaddition, shock/boundary-layer interactionswithin the tip gap lead to large changes in the tip boundary-layer thickness. These effectsgiverisetosignicant differencesintheheat-transfercoefcient withinthetipregioncomparedwiththelowspeedtipow. Comparedwiththelowspeedtipow, thehighspeed tip ow is much less dominated by turbulent dissipation and is thus less sensitive tothechoiceof turbulencemodel. Theseresultsclearlydemonstratethat bladetipheattransfer is a strong function of Mach number, an important implication when consideringthe use of low speed experimental testing and associated CFD validation in engine bladetip design. DOI: 10.1115/1.40024241 IntroductionThedesignofhigh-pressure HPturbinebladetipshasasig-nicant impact on both the aerodynamic loss and the heat load ofunshroudedturbineblades. For thesereasons, therehasbeenasignicant amount ofresearchintothethermouiddynamicsofturbine blade tip ows. Of particular interest has been the accurateexperimental measurement and computational uid dynamicsCFD prediction of blade tip Nusselt numbers since these valuescanbeuseddirectlyintheturbinedesignprocessprovidedthattheowphysicsaresimulatedcorrectly. Insubsonicconditions,this requires matchingtheengine-scaleReynolds numbers. Fortransonicconditionssuchaswhenblade-exitMachnumbersex-ceed 0.9 as what commonly occurs in HP turbines, the effects ofcompressibility become signicant, particularly within the tip gapitself.Asignicant amount of thestudiesintothetipheat transferhaveinvolvedexperimental andcomputational investigationsinlowspeedorsubsonicturbinecascades 1.Recently,lowspeedworkhasshownthatsignicantdifferencesinthepredictedheattransfer canbeobtaineddependingonthechoiceof turbulencemodel 2,3.Far fewer experimental studies have been performed at engineMachnumbersduetothecomplexitiesoftestinginhighspeedows. The water table experiments of Moore et al. 4 and Mooreand Elward 5 showed that the formation of the vena-contracta attheentrancetothetipgapwas abletoacceleratetheowtosupersonicconditionswhenthegapexitMachnumberexceeded0.8. Furthermore, the experimental and numerical tests of Chen etal. 6 on a two-dimensional tip gap in transonic ow showed thatfor an exit Mach number of 1.0, the peak Mach number in the gapwas 1.4. Theyalsoshowedthat whenthe tipowspeedwasincreased from subsonic to supersonic ow, the separation bubbleat the inlet to the gap reduced signicantly in length from 1.5 g toless than 1.0 g.Early tip heat-transfer measurements in high speed rotating fa-cilities were performed by Metzger et al. 7 and Dunn and Kim8. These tests were performed at stage pressure ratios of 2.3 and1.7, respectively, and the blade-exit Mach numbers are estimatedto have been in the region of 0.70.8 9. Later heat-transfer mea-surementsonaattipwerepresentedbyMolteretal. 10atahigher stagepressureratioof around4. Theheat transfer wasmeasuredat four locations onthetipandwas comparedwithCFD. In this case, the blade ow eld was transonic and showedsignicant differences from previous subsonic HP turbine results,such as those of Ameri and Bunker 11.Thorpe et al. 12 took high resolution measurements of the tipheat transfer alongthecamber lineof aHPturbinebladeinatransonic turbine facilityusinganarrayof 17thinlmheat-transfer gauges. In this case, the stage pressure ratio was 3.12, andthe blade-exit Mach number was around 1.0. These results providea detailed data set, which will be used in this paper for validationpurposes. Othertipheat-transfermeasurementsintransonictur-binestages arepresentedbyDidier et al. 13 andChanaandJones 14.Typically, single stage HP turbine engine blade-exit Mach num-bersareintheregionof M=0.91.1, withcorrespondingpeaksuction surface values of M=1.11.3, and thus the peak tip Machnumbers will exceed this. It will be shown in this paper that evenhigh subsonic exit blading M0.8, which does not reach sonicconditionsonthebladesurface, mayhaveregionsof transonicowwithinthetipgapitself. Despitethis, littlehasbeenpub-lishedontheeffect of compressibilityonthetipheat transfer.Therefore, thispaper aimstodeterminethekeydifferencesbe-tweenhighandlowspeedtip-gapowsandwhat effectsthesedifferences have on the tip heat transfer for a at blade tip.In this paper, high speed and low speed tip ows are compared.Fully turbulent CFD predictions are initially validated against ex-perimental data, whichshowgoodagreement withthemeasure-mentsinregionsofturbulentowovertheaftportionofthetip1Corresponding author.Contributed by the International Gas Turbine Institute IGTI of ASME for pub-lication in the JOURNALOF TURBOMACHINERY. Manuscript received February 11, 2010;nal manuscript revised February 22, 2010; published online April 26, 2011. Editor:David Wisler.Journal of Turbomachinery OCTOBER 2011, Vol. 133 / 041025-1 Copyright 2011 by ASMEDownloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termsand blade surface. The fully turbulent CFD overpredicts the heattransfer in regions of laminar and transitional ows, which occurneartheleadingedgeofthetipandbladesurface. Threeturbinebladeprolesarethentestednumerically. Thethreebladesaredesigned to have the same loading distribution and Reynoldsnumber but different exit Machnumbers0.98, 0.67, and0.1.Both the standard k- and SpalartAllmaras turbulence models areused. ComparisonsofthepredictedtipheattransferatthethreeMachnumbers showasignicant dropintipheat loadas theMachnumberisincreased. Thisdropisdrivenbythetransonicnature of the tip ow. The results indicate that the high speed tipowismuchlessdominatedbyturbulent dissipationcomparedwith the low speed ow. Thus, an improved picture for turbine tipows is presented.2 Simulating a Transonic Blade Tip FlowThe3Dcomputationalmeshesusedinthisinvestigationwerecreated using the PADRAMRolls-Royce in-house meshing pro-gram. In order to compare the numerical scheme with experimen-tal data, a 1 and 1/2 stage mesh was used, as shown in the upperpart of Fig. 1. The actual blade tip corner radius was modeled, andthe calculation was run using measured boundary conditions. Thehigh-pressure blade HPB mesh featured 2.5106cells, with 52gridlinesinthetipgapandy+valuesat therst cell fromthesurface of approximately 30 on the blade tip and casing with wallfunctions. Typical gridsizesforthesinglerowstudy showninthe lower part of Fig. 1 were over 3106cells per blade passage,again with y+ values of approximately 2030 when wall functionswere implemented. A quasi-three-dimensional tip model was alsotested numerically, and the mesh for this was created using a codewritten inMATLAB; this mesh is shown in Fig. 2.The Rolls-Royce in-house CFD code, HYDRA, was used to solvetheow. ThisprogramisabletosolvethesteadyandunsteadyReynolds-averaged NavierStokes equations in three dimensions.For the current investigation, steady calculations were performed.The viscous surfaces were isothermal in all cases.TheCFDcalculationsusedfullyturbulentimplementationsofboth the SpalartAllmaras and the k- turbulence models. Assuch, noattempt was madetomodel transitional ow. Recentwork has shown that the standard k- turbulence model gives thebest matchwithlowspeedexperimental data3; however, thisworkalsoshowedthat thek-model gavequalitativelysimilarresults to the k-model. The HYDRAcode with the SpalartAllmaras model has beenusedpreviouslytostudytipleakageows from high speed rig tests by Willer et al. 15, who studiedthetiploss, andby Atkinset al. 16, whostudiedunsteadytipheat transfer. Both investigations found good agreement with ex-perimental results.In order to test the ability of the CFD code to predict blade heattransfer, a series of calculations was compared with previous ex-perimental data, as discussed next.2.1 Transonic LinearCascade Blade Surface Heat FluxComparison. First, a comparison with the experiments of Doorly17 and Nicholson 18 was performed their data are also avail-able in Ref. 19. Both Doorly 17 and Nicholson 18 tested themidspanoweldofatransonicturbinecascade seeTable1.TheexperimentallymeasuredisentropicMachnumber distribu-tionofNicholson 18isshowninFig.3.SeveralCFDcalcula-tions arealsoshown; thesehavedifferent turbulencemodelingand mesh densities. Two SpalartAllmaras calculations are shownin the gure, one with wall functions, where y+ 20 denoted asSAWF, and the other where the laminar sublayer is fully resolvedy+ 1denotedasSA. Ak-calculationwithwall functions,again y+ 20 is also shown denoted as KEWF. The predictionsagreewell withtheexperiment. TheSAcalculationwithfullyFig.1 1and1/2stagemesh topandtypicalcascadebladepassage mesh bottomFig. 2 Quasi-three-dimensional tip model meshTable 1 Details of RT27a cascadeMexit0.96Reexit=VexitCx/ L1.55106in42.75 degout68 degTw/ Tg1.5Pitch-to-chord ratio 0.86Fig. 3 Isentropic Mach number distributions at midspan, com-parison of CFD and experiment041025-2 / Vol. 133, OCTOBER 2011 Transactions of the ASMEDownloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termsresolved boundary layers gives the closest agreement with experi-mentinthediffusingpartofthebladesuctionsurface. Overtheremainderofthebladesurface,thereisverylittledifferencebe-tween the CFD calculations.TheexperimentallymeasuredNusselt numberdistributionsofDoorly 17 and Nicholson 18 are shown in Fig. 4. The experi-ments were performed at high turbulence intensity Tu=5% andlow turbulence intensity Tu0.2%. The turbulence intensity hasasignicanteffectontheNusseltnumberonthesuctionsurfacebecause at low turbulence levels the boundary layer is laminar forapproximately 40% of the suction surface length. The higher tur-bulencelevel signicantlyincreasesthesuctionsurfaceNusseltnumber by promoting transition nearer the leading edge. For thisreason, the CFD calculations with fully turbulent boundary layersSAWF, SA, and KEWF differ signicantly with the experimen-tal dataover theearlysuctionsurface. Thereis amuchbetteragreement over the late suction surface after 50% surface length,wheretransitionismost likelytobecomplete. Alaminar CFDcalculationisalsoshown. Itisimportanttonotethatitwasnotpossible to fully converge this calculation; however, it is interest-ingtonotethat thiscalculationagreeswell withthelowturbu-lence level results of Doorly 17 on the early suction surface. Onthe pressure surface, the effect of turbulence intensity is less pro-nounced.ThefullyturbulentCFDcalculationstendtolieintherange of the experimental data.The results show that in regions of fully turbulent ow, both theSpalartAllmaras and the k- turbulence model give a reasonableagreementwithexperiment. Inregionsoflaminarortransitionalow, the fully turbulent CFD clearly overpredicts the heat transfersignicantly as might be expected.2.2 Transonic Turbine Stage Blade Tip Heat FluxComparison. Inorder totest theveracityof thecomputationalcode for tip heat transfer on a transonic blade, stage calculationsof the Oxford Rotor Facility ORF geometry were performed andcomparedwiththeexperimentaldatapresentedbyThorpeetal.12. The working section of the ORF contains a 0.55 m diametershroudlesshigh-pressureturbinestageandallowsthesimulationofenginerepresentativeMachandReynoldsnumbersaswellastheappropriategas-to-wall temperatureratio. Thereare36inletnozzle guide vanes upstreamof the rotor disk, which has 60bladesandrotatesat 8910rpm. Therotorexit Machnumberis0.98, and the exit Reynolds number based on axial chord is 1.55106. Thestatictipclearanceis2.5%ofthebladeheight. Thestageinlet total temperaturetobladesurfacetemperatureis1.3.Themidspanbladeproleisthesameasthat testedbyDoorly17 and Nicholson 18 discussed previously.SteadyCFDcalculationswereperformedwithamixing-planetreatment between blade rows. Thus, the circumferential averagedpotential eld effects of the upstream and downstream vanes andtheeffectofrelativecasingmotionweresimulatedinthesecal-culations. The tip edge radius was modeled in the CFD geometryin order to match the experimental tip geometry, as described byAtkins et al. 16.Figure5shows acomparisonof theCFDSAWF withthemeasured tip heat ux at the gauge locations of Thorpe et al. 12,which are shown in the lower part of the gure. The experimentaldatashowafallingtrendofheatloadfromgaugelocations13620%axial chord, thenrisinguntil amaximumat gauge650%axial chord, andthenfallingfromgauges713 5090%axial chord. At the location of gauge 1, the turbulent calculationpredictsaheat uxtowithinafewpercent oftheexperimentaldata. However, moving rearward along the camber line, the CFDoverpredicts theheat ux. Fromgaugelocation6onward, theCFD prediction falls and follows both the trend and absolute levelof the experiments very well, to within a few percent at locations7and8, andtoapproximately10%ofthelocal measuredlevelthrough to gauge 12 6%of the maximum measured value.In short, the CFD gives a good match with experiment over themidandaft portions of the tip, while it overpredicts the heattransferneartheleadingedge. Inordertounderstandwhy, it isFig.4 Nusseltnumberdistributionsatmidspan,comparisonof CFD and experimentFig. 5 Comparisonof experimental andpredictedbladetipheatuxwiththemappingbetweenexperimentalgaugeloca-tions and the CFD meshJournal of Turbomachinery OCTOBER 2011, Vol. 133 / 041025-3Downloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termsusefultoinvestigatefurtherthedifferencesbetweenthetipownear the leading edge and the ow over the aft portion of the tip,as described next.2.2.1 NeartheLeadingEdge. Theowthat passesovertherstfourgaugelocationscomesfromneartheleadingedge seeFig. 6. Unlikethemajorityofthetipow, thisboundary-layerowdoesnotseparateasitentersthetipgapbutratherexperi-encesalongregionofacceleration.Furthermore,thestreamwiseReynolds numbers at gauges 14, based on the local ow condi-tions, arebelowthecritical valuefortransitional owonaatplatewithazeropressuregradientReL3105. Consideringthat the leading edge of the tip region is subject to the ingestion ofacomplexshearedcasingboundarylayer, as well as unsteadywakepassingevents, atransitional boundarylayer istobeex-pectedinthisregion theseunsteadyeffectsarenotsimulatedinthe CFD with a mixing-plane treatment. To illustrate this, a lami-narsolutionhasbeenobtained usingthefullyconvergedturbu-lent predictionasaninitial conditiontoestimatetheheat ux,whichwouldresult fromafullylaminarboundarylayerinthisregion. It should be noted that the solution is not fully convergedandprovidesanestimateonly. ThelaminarvaluesareplottedinFig. 5 and show excellent qualitative agreement with the trend ofthe experimental data at the rst four gauge locations, albeit withan absolute value of approximately 50% of the measured values.This is a strong indication that the tip boundary layer close to theleading edge is transitional rather than fully turbulent, as was alsoobserved on the blade suction surface.2.2.2 Mid Chord and Aft. Figure 6 shows the computationallypredicted blade tip surface heat ux and isentropic Mach numbercontours. Mach number contours are also shown on planes paral-lel to the tip ow over the mid and aft portions of the tip. In thisregion, the tip surface boundary layer separates from the pressuresideedgeof thetip, causingaregionof lowheat ux. Thisisfollowedbyareattachment indicatedbyawhitedashedline,which causes a rise in the heat transfer.The ow can be divided into a region where this reattachmentoccurs in subsonic ow region Y, over the mid portion of the tip,andtheregionwherethereattachmentoccursinsupersonicowseparation, over the aft portion of the tip region X.InregionY,wherethereisasubsonicreattachment,thereisahigh level of heat ux downstream of the reattachment. The peakheat uxisroughly50%higher thanthetiparea-averageheatux. In addition, the separation length is large.In region X, where there is a supersonic reattachment, the sepa-rationlengthreducessignicantlycomparedwiththeregionofsubsonic reattachment. The Mach number contours in this regionshowthat theowaccelerates over theseparationbubbletoaMachnumber of over 1.5. Theowremainssupersonicdown-stream of the bubble and then decelerates across a normal shock attheexitofthetipgap. Thelevelofheatuxdownstreamofthebubble is signicantly lower than over the mid portion of the tip.The reasons for this will be shown later.3 High Speed Versus Low Speed Tip FlowInordertocomparethetransonictipowwithananalogouslowspeedtipow, itwasnecessarytodesignlowspeedbladesthat matchedthechord-to-gapratio, thebladeloading, andtheReynolds number of thetransonicblade. Thetipsection95%span of theOxfordrotor bladewasusedasthebaselinehighspeedgeometrybladeA, whichhasanexit Machnumber ofMexit=0.98. The predicted Mach number distribution for this canbe seen in Fig. 7. The peak Mach number is 1.15, and the ow issupersonicbetweenapproximately25%and75%ofthesuctionsurface length.Figure 7 also shows the isentropic Mach number distribution oftwoother bladeproles: bladeBwithMexit=0.67andbladeCwith Mexit=0.1 typical low speed blade, which were designed tomatchthebladeloadingandReynoldsnumber of thetransonicbladeA. Thebladeproles canbeseeninFig. 8. Ingeneral,reducing the Mach number requires an increase in blade thickness.Therefore, a further constraint on the prole design was made soFig. 6 Predictedheat uxandMachnumber contoursandstreamlines SpalartAllmaras+WFFig.7 IsentropicMachnumberdistributionsforblades A,B,and CFig.8 PredictedtipisentropicMachnumberforblades A,B,and C normalized by exit Mach number041025-4 / Vol. 133, OCTOBER 2011 Transactions of the ASMEDownloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termsthat thetrailing-edgethickness-to-chordratiowasheldconstant.This ensured that the trailing-edge loss would not increase signi-cantly. Figure7showsthat thesuctionsurfacedistributionsarematchedwell. Thereisapoorermatchonthepressuresurface,which was a consequence of maintaining a constant trailing-edgethickness-to-chord ratio while matching the suction surface veloc-ity distribution.3.1 Effect of Mach Number on Tip Heat Flux.Inordertostudytheeffect ofMachnumberonthetipheat transfer, three-dimensional CFD calculations of the blade proles with a tip gap-to-axial-chord ratio of 5% were performed. This matched the Ox-fordrotorbladedescribedpreviously.Thewalltemperaturewasspeciedinthesecalculations sothat Tg/ Tw=1.5. CalculationswereperformedwithbothSpalartAllmarasandk-turbulencemodels with wall functions. This helps to eliminate effects due tothechoiceofturbulencemodel sincethishasbeenshowntobeimportant at lowspeedconditionsinpreviousstudies 2,3. Forthese calculations, both the blade and the casing wall werestationary.The predicted tip surface isentropic Mach number contours forblades A, B, and C are shown in Fig. 8. The black lines indicatecontours of M=1.0. The gure shows that much of the blade A tipis supersonicwiththepeakMachnumber exceeding1.5. Thisoccursovertheseparationbubblenearthepressuresurfaceedgeof the tip. For blade B, there is a small supersonic patch, but themajority of the tip is subsonic. For blade C, the ow is predomi-nantly incompressible with the Mach number never exceeding 0.2.Figure8showsthat thetipsurfacepressuredistributionsaresimilarforthethreebladeproles. Themaindifferencesinthepressuredistributionsoccurnearthepressuresideedgeandareduetochangesintheseparationbubblegeometry.Thereisverylittle difference between the pressure distributions for the two tur-bulencemodelsonthehighspeedblade Mexit=0.98.However,for thelowspeedbladeMexit=0.1, thepressuredistributionsdiffermoresignicantly, andagainthisismainlyduetodiffer-ences in the prediction of the pressure side edge separationbubble.Figure 9 shows the predicted tip surface heat ux contours forthethreebladeproles.Forthefrontportionofthetip,between0%and10%axial chords, theheat uxremainsrelativelyun-changed for the three blades. However, over the remainder of thetip, there is a marked increase in the heat ux as the Mach numberis reduced. The overall changes in tip heat load can be seen in Fig.10. Reducing the Mach number from 0.98 to 0.1 increases the heatload by 60%.It is interesting to note that the difference in predicted heat loadbetween the KEWF calculation and the SAWF, as a proportion oftheir mean, does not rise at higher Mach numbers. In fact, there isa drop in this difference from24%at M=0.1 to 20%at M=0.98 see Fig. 10. This indicates that the high speed tip ow isless sensitive to the choice of turbulence model compared with thelow speed tip ow, as was also observed in the tip Mach numberdistributions shown in Fig. 8.Inorder tounderstandfurther thechangeinheat loadwithMach number, it is useful to consider that the surface heat ux isgiven byq = hTaw Tw 1At lowMachnumbers, thedrivingtemperaturedifferenceisconstant overthetipsurface. Thisisbecausetheadiabaticwalltemperature is given by the gas total temperature, which does notvarysignicantlyoverthetipsurface. However, foratransonictip, the adiabatic wall temperature is given by the local recoverytemperature, which is a function of local Mach number and recov-ery factor. Therefore, both the driving temperature difference andthe heat-transfer coefcient must be considered.Thedriver temperatureandheat-transfer coefcient arebothshowninFig. 11forblade A. ThesehavebeencalculatedfromCFD calculations at two different wall temperatures: at the base-linewall temperature, Tw2=2/ 3Tgandtheincreasedwall tem-peratureofTw1=Tg. Thus,theheat-transfercoefcientandadia-batic wall temperature were given by h=q1q2 / Tw2Tw1 andTaw=q1/ q2Tw2Tw1 / q1/ q2 1.Figure11showssignicant spatial variationsintheadiabaticwall temperature, which give rise to a maximum drop in the drivertemperature of over 25% in the region of the pressure side edge ofthetipwhenTg/ Tw=1.5. Thisoccursintheregionof peaktipMachnumberovertheseparationbubble seeFig. 8. Thispar-Fig. 9 Predictedtipheat uxforbladesA, B, andC Tg/ Tw=1.5Fig. 10 Predicted variation in area-averaged tip heat loadFig. 11 Predicted adiabatic wall temperature and Nusselt num-ber for blade A Mexit=0.98, Tg/ Tw=1.5Journal of Turbomachinery OCTOBER 2011, Vol. 133 / 041025-5Downloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termstiallyexplainswhythelowerspeedblades bladesBandCex-hibit high area-average heat loads, although this contributes a rela-tively small amount.Themost signicant causeof theincreasedheat uxat lowspeeds is due to an increase in heat-transfer coefcient. As shownin Fig. 10, the change in average recovery temperature hTwTrec accounts for only 7%of the increased area-averaged heat load as Mexit is reduced from 0.98 to 0.1, while thechangeinheat-transfer coefcient hTwTrec contributes93% to this change when Tg/ Tw=1.5. It is important to note thatthe effect of changes in recovery temperature on heat load is de-pendent on Tg/ Tw. For smaller Tg/ Tw, changes in recovery tem-perature will affect the heat load more signicantly.Themainreasonfor theincreasedheat-transfer coefcient atlowspeeds canbeseeninFig. 12, whichshows theturbulentviscosityratioonacutplaneat4%ofthetipgapabovethetipsurface. The gure shows that for blade C, the turbulent viscosityin the tip gap is roughly twice that of blade A between 30% and100% axial chords. This increased turbulence will tend to increaseboth the momentum and heat transfer within the tip, thus increas-ing the tip heat-transfer coefcients. Figure 12 also shows that theKEWFcalculations giveahigher turbulent viscositycomparedwith the SAWF calculations for both blades A and C note that thecolor scale for KEWF is 2.7 times larger than SAWF. This causesthe increased heat ux in the KEWF calculations observed earlierin Fig. 9.The generation of turbulence in the tip gap is highly dependentonthenatureof thebubblereattachment. Thestructureof theseparation bubble for blades A and C can be observed in Fig. 13.This gure shows the Mach number contours on a cut plane par-allel to the ow indicated in the gure. It can be seen in Fig. 13that thereattachment onbladeA Mexit=0.98 occurspredomi-nantlyinasupersonicow, whichmust alsolocallyaccelerate,while in blade C Mexit=0.1the reattachment occurs in a regionof pressure recovery brought about by turbulent mixing.Figure 13 shows the predicted heat transfer in this region fromthe SAWF calculations. For the high speed blade A Mexit=0.98,the rapid thinning of the accelerating boundary-layer downstreamof the bubble leads to an increased heat-transfer coefcient. Thisisterminatedbyarapidthickeningoftheboundarylayerduetotheinteractionofthenormal shock, whichcausesadropintheheat-transfer coefcient. Thus, the strong pressure gradients in thehigh speed tip ow give rise to signicant variations in the localheat transfer, which are not observed in the low speed tip ow.Figure 13 also highlights that the tip width-to-gap ratio w/g ofthe low speed Blade C is greater than Blade A. The variations ofw/g for both blades A and C are shown in Fig. 14. The variationsof w/g for both blades A and C have been computed based on thetipstreamlines,ratherthantheactualbladewidth,sinceitisthestreamwise tip width which is of most importance to the tip ow.Blade C has a 2030% higher width-to-gap ratio between 030%axial chord, anda50100%higher width-to-gapratiobetween50% and 70% axial chords. Therefore, the effects of the differenceinwidth-to-gapratioarelikelytobemost signicant over thelatter portion of the tip.The following section uses a quasi-three-dimensional model ofthetipowtoinvestigatetheeffectsofbothwidth-to-gapratioand Mach number on the tip ow.4 Quasi-Three-Dimensional Tip FlowInordertoinvestigatetheeffectsofMachnumberandwidth-to-gapratioonthebubblereattachment, quasi-three-dimensionalCFDcalculationsof agapowwereperformed. Thecomputa-tional domain and mesh are shown in Fig. 2. A baseline width-to-gapratioofw/ g=5.0waschosen,andthetipReynoldsnumberbasedonthetipwidthandtheexit velocitywas Re=4105.These values are typical of the Oxford rotor blade describedabove. ASpalartAllmaras turbulencemodel was usedwithoutwall functions y+ 1.The effect of the width-to-gap ratio on the quasi-three-dimensional gapowcanbeseeninFig. 15. ThegureshowsMachnumber contoursof thepredictedgapowfromcalcula-tions at ve different width-to-gap ratios, with a constant isentro-Fig.12 Predictedturbulenttolaminarviscosityratio T/ Lfor blades A and C, cut at 4% tip gap above tip surfaceFig. 13 Mach number contours on a cut plane through the tipgapandsurfaceheat-transfer contoursfor bladesAandCSAWFFig. 14 Width-to-gap ratio against axial chord blades A and C041025-6 / Vol. 133, OCTOBER 2011 Transactions of the ASMEDownloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termspic exit Mach number of 0.9. For ease of viewing, the gures areplotted so that they appear to have the same aspect ratio; however,thetrueaspect ratiosareindicatedbelowthecontour plots. Atw/ g=1.75, a normal shock exists downstream of the bubble reat-tachment. Doubling the tip width w/ g=3.5, causes the shock tomove upstream as a proportion of the tip width. In addition, theow reaches a signicantly higher peak Mach number of over 1.5.Increasingthetipwidthfurther continues tomovethenormalshock upstream.Betweenw/ g=3.5and14, the qualitative tipowstructuredoes not vary signicantly, and thus the difference in width-to-gapratiobetweenthehighspeedblade A andlowspeedbladeCisunlikely to have a signicant effect on the tip heat transfer.At w/ g=28, thetipseparationisconnedtoroughly10%ofthe tip surface, and the remainder of the gap ow is akin to a fullydevelopedpipeow, wherethetipandcasingboundarylayersextendtollthegap. WhenscaledtothewidestpartoftheOx-ford rotor RT27a blade, the w/ g=28 case corresponds to a gapto span ratio of approximately 0.7%.The effect of compressibilityonthe quasi-three-dimensionalgap ow can be seen in Fig. 16. This gure shows contours of thepredictedMachnumberdistributionwithinthegapataseriesofisentropicexit Machnumbers from0.1to1.0w/ g=5.0. Be-tweenMexit=0.1and0.6, thereisagradual reductioninbubbleheight but very little change in the qualitative nature of the ow.This reduction in bubble height is coupled with an increase in theaccelerationovertheinitial part oftheseparatedshearlayer. AtMexit=0.8, the peak Mach number reaches 1.2, and a normalshockformsdownstreamof thebubble.At Mexit=1.0, thepeakMach number exceeds 1.5, and there is a dramatic reduction in thebubble length.Figure17plotsthebubbleheightandlengthwithMachnum-ber. The bubble height reduces by a factor of 2 when the tip-gapexit Machnumberisincreasedfrom0.1to1.0. Incontrast, thebubblelengthremainsroughlyconstant astheMachnumber isincreaseduntil thereisasupersonicreattachment ofthebubble,which causes the bubble length to reduce suddenly by a factor of2. A similar shortening of the bubble was observed by Chen et al.6, who saw a reduction in bubble length from 1.5 g to less than1gwhentheowspeedwasincreasedfromsubsonictosuper-sonic.Figure16alsoshowsthatforMexit=1.0, obliqueshocksformdownstream of the bubble, which reect from the tip and casing,causingarapidthickeningofbothboundarylayers.Betweentheshock reections, expansion waves cause a signicant thinning oftheboundarylayers. Thesupersonicregionis terminatedbyanormal shock, which causes a further thickening of the boundarylayer. This shockstructureis verysimilar tothat observedbyMoore et al. 4 and Moore and Elward 5.Thevariationinturbulent viscosityinthetipgapastheexitMach number is increased is plotted in Fig. 18. From Mexit=0.1 to0.8,thereisagradualreductionintheproductionofturbulence.Thiscorrespondstotheincreasingaccelerationoftheowovertheseparatedshearlayerneartheinlettothegap,simplyduetotheeffect of compressibility. However, at Mexit=1.0, thereisasuddendropintheturbulenceproductionwhenthebubblereat-tachment occurs in supersonic accelerating ow. Thus, the bubblereattachment is brought about by this accelerating pressure gradi-ent, which causes a signicant foreshortening of the bubble and inturn produces much less turbulent mixing.5 DiscussionThehighspeedtipowstructureis showninFig. 19. Thisgure has been constructed from an inviscid duct calculation us-ingthepredictedbubbledisplacementthicknessfromagapexitMach number of Mexit=1.0 and w/ g=5 see Fig. 16. As the owFig. 15 Effect of width-to-gap ratio on quasi-three-dimensional tip ow Mexit=0.9Fig. 16 Effect of pressure ratio on quasi-three-dimensional tipow w/ g=5Fig. 17 Separationbubbleheightandlengthfromthequasi-three-dimensional tip ow w/ g=5Journal of Turbomachinery OCTOBER 2011, Vol. 133 / 041025-7Downloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termsenters the tip gap, the ow separates from the pressure side corner.The separation accelerates the ow to supersonic like aconverging-diverging nozzle; thus, as the separation begins to re-attach,thechangeinareaforcesthemainstreamowtoacceler-ate. This, in turn, aids the reattachment process, and the separationis closed much earlier than would occur through turbulent mixingalone.Thisextramechanismforsupersonicreattachmentcannotoccur in subsonic ow.Compressionwavesformduetothecurvatureofthereattach-ingboundarylayer seeFig.19. Thesecoalesceintoanobliqueshock, which reects fromthe tip and casing. Between theseshock reections, the ow rapidly accelerates. These rapid accel-erations and decelerations cause large local thinning and thicken-ingofthetipboundarylayer. Thissupersonicowisterminatedbyanormal shock, whichcausesafurtherthickeningofthetipboundary layer.This basic structure occurs over a signicant portion of the tiponboththehighspeedcascadebladeseeFig. 13 andontherotating turbine blade see Fig. 6. In essence, this high speed tipowisdominatedbystronglocal pressuregradients, whilethelowspeedtipowisgovernedbytheoverall pressuregradientacrossthetipwidth. Thiscanbeunderstoodbyconsideringthecombined energy and continuity equations asdVV= dAA1M2 12Thus, for the same change in boundary-layer displacement thick-ness, the mainstream ow will experience a greater acceleration athigher Mach numbers. This acceleration will then reduce the rateofboundary-layergrowthandthusmoderatetheincreaseinac-celeration. This effect was observed in Fig. 16. The gure showedthat astheexit Machnumberwasincreased, fromMexit=0.1to0.8,theaccelerationovertheseparationbubbleincreased,whichin turn gradually reduced the height of the bubble.In the high speed tip ow, Eq. 2 also shows that downstreamofthebubblethroat, thecouplingbetweentheinviscidandvis-cous ow elds is contrary to that in the low speed tip ow. Thisis because while in subsonic ow, the reduction in blockage due tothe boundary-layer reattachment causes a streamwise pressurerise, andinsupersonicowthereductioninblockagecausesapressure drop, which further aids the boundary-layer reattach-ment. At the same time, the increased acceleration of the separatedshear layer and the reduction of bubble length lead to a reductioninturbulenceproductionforthehighspeedtipow. Therefore,thehighspeedtipowismuchmoredominatedbystronglocalpressuregradientsthanturbulent dissipationortheoverall pres-sure difference across the tip.Consideringthe3Dbladetip, withinthesupersonicregions,this acceleration and reduction of turbulence aid the accurate pre-diction of the heat ux. As the ow eld is predominantly invis-cid, thephysicsarewell capturedbytherelativelysimpleoneequationmodel withawall functiontypeapproach; hence, theheat-transfercoefcientiswellpredictedbytheassociatedRey-noldsanalogy.Withinthesubsonicregionsonthebladetip, theheat-transfer coefcient is governed by a complex combination ofsubsonic turbulent reattachment and highly skewed boundary lay-ers, for which the basic ow structure is dependent on the turbu-lence modeling.It is also worth noting that as blade-exit Mach numbers increaseabove 0.8, the results indicate that the blade tip will start to chokewhile the blade mass ow will continue to increase. Consequently,we would expect the tip leakage mass ow rate as a proportion ofthe mainstream ow to reduce at exit Mach numbers greater than0.8, which will tend to reduce losses.6 ConclusionInthispaper, lowspeedandhighspeedtipowswerecom-pared for a high-pressure turbine blade at the same Reynolds num-ber. Athighspeed Mexit=0.98themajorityofthetipowwastransonic, while at low speed Mexit=0.1 the ow was essentiallyincompressible.Theresultsshoweda60%dropintipheatloadforthehighspeedblade. Thiswasduetoacombinationofre-duceddriver temperature andreducedheat-transfer coefcient.High Mach numbers over the pressure side corner separationbubbleledtoa25%reductioninthelocal driver temperaturedifference when Tg/ Tw=1.5. This effect will be amplied atlowergas-to-walltemperatureratios.However,themostsigni-cantcauseofthedropinheattransferwasduetoareductioninheat-transfer coefcient. This occurred in the high speed ow duetoareductioninturbulentmixing.Theturbulencewasfoundtoreduceasaresultofanincreaseintheaccelerationofthemain-streamowovertheseparationbubblecombinedwithahalvingof the bubble length brought about by the supersonic accelerationof the reattaching boundary layer.Theresultsindicatethat forturbinetipowsat engineMachnumbers, the tip ow structure is signicantly different from thatsuggestedbylowspeedcascadetesting. At highspeeds, thetipheat uxisbothquantitativelyandqualitativelydifferent com-paredwiththeincompressibletipow. Fundamentally, thehighspeedtipowismuchmoredominatedbylargelocal pressuregradients and less dependent on turbulent dissipation than the lowspeed tip ow. The results also indicate that the accuracy of CFDFig. 18 Effect of pressureratioonturbulent viscosity w/ g=5Fig. 19 Transonic tip ow Mach contours041025-8 / Vol. 133, OCTOBER 2011 Transactions of the ASMEDownloaded From: http://turbomachinery.asmedigitalcollection.asme.org/ on 12/06/2014 Terms of Use: http://asme.org/termspredictions for highspeedtipheat transfer is likelytobelessdependent onthechoiceof turbulencemodel thanthat for lowspeed tip ows.AcknowledgmentThe authors would like to thank Rolls-Royce plc for their sup-port and provision of the CFD and meshing code.NomenclatureAareaCxaxial chordg tip-gap heighth heat-transfer coefcientk conductivity at wall temperaturekawconductivity at adiabatic wall temperatureLseparation bubble lengthM, MisenMach number, isentropic Mach numberNu Nusselt number=hCx/ kq surface heat uxRe Reynolds numbers surface distance from leading edgeS0total surface lengthTawadiabatic wall temperature, recoverytemperatureTo, Tggas stagnation temperatureTu turbulence intensityTwwall temperatureVvelocitywtip width parallel to tip owy+nondimensional wall distanceswirl angleseparation bubble displacement thickness densityLlaminar viscosityTturbulent viscosityReferences1 Bunker, R. 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