Aerothermal Investigation of Backface Clearance Flow in Deeply Scalloped Radial Turbines

Ping He Graduate University of ChineseAcademy of Sciences,Beijing,100190,China; Institute of Engineering Thermophysics, Chinese Academy of Sciences,Beijing,100190,China e-mail:heping@Zhigang Sun e-mail:sunsonofjilin@Baoting Guo e-mail:guobt@Haisheng Chen e-mail:chen_hs@Chunqing Tane-mail:tan@ Institute of Engineering Thermophysics, Chinese Academy of Sciences,Beijing,100190,China Aerothermal Investigation of Backface Clearance Flow in Deeply Scalloped Radial TurbinesA numerical investigation offlow structure and heat transfer in the backface clearance of deeply scalloped radial turbines is conducted in this paper.It is found that the leakage flow is very strong in the upper radial region whereas in the lower radial region,the scrapingflow dominates over the clearance and a recirculation zone is formed.Pressure distributions are given to explain theflow structure in the backface clearance,and it is found that due to the sharp reduction of radial velocity and Coriolis force,the pressure difference in the lower radial region is reduced drastically,which is the mechanism for the domination of the scrapingflow and the corresponding recirculation zone.There are two high heat transfer coefficient zones on the backface surface.One is located in the upper radial region due to the reattachment of the leakageflow and the other is located in the lower radial region caused by the impingement of the scrapingflow.Increase of the clearance height reduces the high heat transfer coefficient caused by the impingement of the scrapingflow,although it increases the leakage loss.On the other hand,the high heat transfer coefficient in the upper radial region can be reduced remarkably by using the suction side squealer geometry.[DOI:10.1115/1.4006664]1IntroductionRadial turbines have established prominence in thefield of small turbomachinery because of their simplicity,low cost,rela-tively high performance,and favorable installation features[1]. Deeply scalloped radial turbine rotors"scallop"their disk to reduce weight and minimize centrifugal stress.Meanwhile,the highest stress in deeply scalloped radial rotors occurs at locations where the metal temperature is considerably lower than at rotor inlet[2].With these significant advantages,deeply scalloped ra-dial turbines are widely used in microturbines and small turboma-chinery;examples are the radial turbines for the Capstone production series,the NASA high temperature radial turbine for a small engine[3],and the NASA-PW compact radial turbine[4]. As shown in Fig.1,the disk of deeply scalloped radial turbine rotors does not extend out to the blade tip,and the blades have both the tip clearance and the backface clearance in the inlet sec-tion(Fig.2).The backface clearance in deeply scalloped radial turbines causes two additional problems compared with traditional ones.Firstly,the backface clearance leakageflow decreases the blade loading and interacts with the mainstream and secondary flows in the blade passage,which result in lower turbine effi-ciency.Secondly,the blade backface is subjected to relatively high thermal loads due to the high velocity,thin boundary layer, and high temperature associated with the backface leakageflow. The backface leakageflow separates and reattaches in the clear-ance,which leads to a more complicatedflowfield.What is worse,the root of the blade backface is one of the life-limiting regions for radial turbine rotors because of the high centrifugal stress[1],which makes the high thermal loads of the blade back-face more undesirable(in contrast,the blade tip is also subjected to high thermal loads,while the centrifugal stress is extremely low).These two factors greatly affect the efficiency and durability of radial turbines.The backface clearance is usually the largest clearance in deeply scalloped radial turbines,hence,the corresponding loss is considerable.Simonyi et al.[5]presented a comparison of a quasi-3D analysis and experimental performance for three deeply scalloped radial turbines.A hub clearance model was used to eval-uate the influence of the backface clearance.It was found that the efficiency penalty due to the backface clearance was nearly60% of the total clearance losses.A similar conclusion was also drawn by Tirres[6]when he predicted the performance of the solid ver-sion of a cooled radial rotor,and it was shown that the backface clearance loss for the rotor accounted for79%of total clearance losses.Rogo[7]summarized the clearance losses in radial tur-bines as a function of normalized clearance height,the turbine ef-ficiency decreased by0.17%with increasing the normalized backface clearance height by1%.However,there is little literature concerning the detailedflow structure in the backface clearance.Zangeneh et al.[8]analyzed theflow structure in radial turbines.The authors derivedan Fig.1Deeply scalloped radial turbine rotors:(a)turbine A and (b)turbine BContributed by the International Gas Turbine Institute(IGTI)of ASME for publi-cation in the J OURNAL OF T URBOMACHINERY.Manuscript received July28,2011;final manuscript received December4,2011;published online October31,2012.Assoc. Editor:Ricardo F.Martinez-Botas.expression for the generation of vorticity in a rotating coordinate. The blade passage was then divided into three sections according to the expression and three different types of secondaryflows were shown in the rotor.Heidmann et al.[9]investigated the blade surface pressure and outletflowfield of a deeply scalloped radial turbine.The authors found that the simulation results with backface clearance agreed better with the experimental data.A similar result was also found by Larosiliere[10],who used a back-face clearance model to simulate the backface clearanceflow in a deeply scalloped radial turbine.The author presented entropy con-tours and velocity vectors at different quasi-normal planes,and showed the backface clearance leakage vortex.Nevertheless,the flow structure in the backface clearance cannot be simulated by using a clearance model.Cox et al.[11]studied theflow around the scallops of a mixed-flow turbine for turbochargers;the authors indicated that the presence of scallops introduced a forward-facing step to the inletflow,which resulted in a stronger passage vortex and lower efficiency.But the turbine was just scalloped a little,and the investigated region was more like a backface cavity instead of a backface clearance.A numerical study was performed by He et al.[12],who investigated theflow and loss in deeply scalloped radial turbines.Meshes were generated in the backface clearance region to study the backface clearanceflows.It was illustrated that the backface clearance leakage vortexflowed towards the blade tip under the influence of spanwise secondary flow and centrifugal force,which resulted in a large mixing loss. However,the study focused on the backface clearanceflow in the blade passage and theflow structure in the backface clearance was not further discussed.Heat transfer on turbine blade clearances has been a hot topic over the past decades,and most of the works focused on the tip clearance in axial turbines.Bunker[13]presented a detailed review of both numerical and experimental researches on turbine blade tip heat transfer from1955to2000.An early experimental study on blade tip heat transfer conducted by Mayle et al.[14]indicated that the heat transfer coefficient on the blade tip was mainly determined by the pressure-driven tip clearanceflow and was independent of the relative motion between blade and casing wall.Heat transfer on a typical blade tip of power generation gas turbines was experimen-tally studied by Bunker et al.[15],and a complex distribution of heat transfer on the blade tip was observed.There was a sweet spot of low heat transfer that extended into the midchord region within the thickest portion of the tip.And a pressure side entry vortex caused a significant enhancement to heat transfer aft of the sweet spot.Detailed heat transfer behavior of both theflat and squealer tips was experimentally studied by Azad et al.[16,17].It was found that the heat transfer coefficient on the blade tip increased with increasing the clearance height and turbulence intensity,and the overall heat transfer on the blade tip was reduced by using squealer tips.Kwak et al.[18]measured the heat transfer coefficient on the blade tip,shroud,and pressure and suction surfaces for different squealer geometries.It was indicated that the squealer tips signifi-cantly reduced the overall heat transfer and the suction side squealer tip had the lowest heat transfer coefficient among the squealer geometries.Heat transfer coefficient measurements on the blade tip and near tip region conducted by Newton et al.[19] revealed that the highest levels of heat transfer were located where theflow reattached on the tip surface,and high heat transfer was also found on the suction surface,onto which the tip leakage vortex impinged.Heat transfer of a full-scale rotating turbine tip was measured by Dunn et al.[20].The highest Nusselt number was measured just aft of the leading edge lip on the tip cavity surface due to theflow reattachment.Numerical investigations play an important role in the study of heat transfer by providing detailedflow structure and heat transfer results.Mumic et al.[21]presented a comparison of simulation results and experimental data[16,17]of heat transfer on blade tips. The predicted heat transfer for both theflat and squealer tips agreed well with the experimental data.Krishnababu et al.[22,23]per-formed numerical studies offlow and heat transfer on blade tips. The authors presented complex separation and reattachment of leak-age vortex to explain the distribution of heat transfer coefficient in the clearance.It was also shown that the calculation results using the shear stress transport(SST)turbulence model was in close agree-ment with the experimental data[19].Molter et al.[24]presented a comparison of simulation and measurement results in a rotating tur-bine tip.The authors indicated that the computationalfluid dynamics (CFD)calculation did capture the measured heatflux trends in the tip region,although it generally overpredicted the overall heatflux. There were many other studies focused on the heat transfer in axial turbine blade tips,and the detailed review can be seen in Ref.[25]. However,there is little literature concerning the detailed heat transfer in the backface clearance of deeply scalloped radial tur-bines.Heuer et al.[26,27]conducted a comprehensive investiga-tion on the heat transfer of a turbocharger using a conjugate heat transfer method.The authors presented detailed temperature and Nusselt number distributions on the blades.The thermal and cen-trifugal stresses at steady and transient states were also analyzed. It provides a good reference for the current study.This paper presents a numerical investigation offlow structure and heat transfer in the backface clearance of deeply scalloped ra-dial turbines.And the influences of clearance height and squealer geometry are also discussed.2Numerical MethodThree turbines(turbine A,turbine B,and turbine C)are ana-lyzed in this paper.Turbine A(Fig.1(a))is for detailed study;tur-bine B(Fig.1(b))and turbine C are for CFD verification.Turbine A is a deeply scalloped radial turbine designed for a microturbine, turbine B is a deeply scalloped radial turbine designed by PW Company for a small engine[4],and turbine C[16]is a linear cas-cade with the blade tip profile of GE-E3gas turbine rotor blades. The parameters for turbine A and B are shown in Table1. Figure2presents the meridional view for turbine A and both the tip and backface clearances are shown.In Fig.2,the interface of the backface clearance and the backface cavity regions,which are located near the region of RL¼0%,is assumed to be a“wall.”This is often true in experimental conditions,where the sealing of the turbine backface is perfect and the backface cavityflow rate is ignored.However,in real engine conditions,sealingflows($1% Fig.2Meridional view of theflow passage for turbine Aof the mainstream mass flow [7])must be channeled from the compressor to prevent hot gas ingestion.This paper focuses on the nonsealing flow condition and the influence of the sealing flow on the backface clearance flow,especially near the region of RL ¼0%,will be the next step of the study.The squealer tip is widely used in axial turbines,and it can reduce the thermal loads for the turbine blade tip as mentioned in the above references.However,the effect of the possible squealer geometries on the heat transfer of blade backface surface is not yet clear.Figure 3shows the three squealer geometries for the blade backface in this paper.They are pressure side squealer (PSS),suction side squealer (SSS),and pressure and suction side squealer (PS þSS),respectively.Structured meshes are generated using ANSYS-ICEM,a com-mercial mesh generation software.The computational domain con-sists of a single stator and rotor for turbine A and B and a single blade for turbine C with periodic boundaries.Meshes are generated in the backface clearance region to obtain the detailed flow struc-ture where the average Y þis less than 1.To ensure mesh independ-ence,the flat backface case requires $1.9Â106cells,the squealer backface case $3.8Â106cells for turbine A.The detailed cell sta-tistics are summarized in Table 2and the meshes are shown in Fig.4(a coarse mesh is used for better visualization).A commercial solver software,ANSYS-CFX,is used to per-form the calculations,and the solutions are obtained by solving Navier–Stokes equations.For turbine A and B,total pressure and total temperature are specified at the stator inlet along with a ra-dial flow angle (0deg)and medium turbulence intensity (5%).At the rotor outlet,static pressure is specified.A rotational periodic boundary condition is imposed on the periodic boundaries.The connection option of the stator and rotor domains is “stage,”which performs a circumferential averaging of the fluxes through bands on the stator and rotor domain interface.The isothermalwall condition is imposed on the blade surfaces with 75%of inlet total temperature for the calculation of heat transfer coefficients.Although the heat transfer coefficient may vary with different given wall temperatures,the distribution trends are the same.And the constant temperature wall condition was widely used in the calculation of heat transfer for tip clearance [21–24,28].ATable 1Parameters for the two turbinesTurbine ATurbine B Tt 0,K 1150477.8Pt 0,kPa375.6333.7Mass flow,kg/s 2.08 2.71Rotor speed,rpm 4500019919Pt 0/Pt 2 3.51 4.99Ps 2,kPa9652Work factor (D h/Ut 2)0.92 1.10Rt ,mm 125183.6Rs/Rt0.640.68Blade tip passage span,mm 2013.4Tip clearance at inlet,%64Tip clearance at outlet,%0.80.46BC ,%8 2.5Stator number/Rotor number23/1336/14Fig.3Different squealer geometries for turbine A:(a )PSS,(b )SSS,and (c )PS 1SSTable 2Cell statistics for the three turbines Turbine A (Flat)Turbine A (SSS)Turbine B Turbine C Stator 0.6M 0.6M 0.6M —Rotor 1.3M 3.2M 1.3M 1.5M Total1.9M3.8M1.9M1.5MFig.4Meshes for the three turbines:(a )turbine A,(b )turbine B,(c )the rotor of turbine A with SSS backface,and (d )the blade of turbine C with SSS tipcounter-rotating wall condition is specified for the backface and tip casings.In this case,the backface casing is stationary in the absolute coordinate system,whereas in the rotating coordinate system,it moves with an angular velocity(x)in the counter-rotating direction and the relative casing motion to the blade is simulated.For turbine C,total pressure(126.9kPa)and total tem-perature(297K)are specified at the blade inlet.The inletflow angle is32deg and the turbulence intensity is7.9%.At the bladeoutlet,static pressure(102.7kPa)is specified.A constant wall temperature condition(328K)is imposed on the blade tip accord-ing to the experiment.The low Reynolds number SST k-x turbu-lence model,which gives highly accurate prediction of theflow separation,is used during the simulation.And it successfully cap-tured the heat transfer features in tip clearance as mentioned in References[21–23].3Results and Discussion3.1CFD Verification.The measurement of the clearance flow is a challenging task.And there is little experimental data on theflow structure or heat transfer of backface clearance for refer-ence.Therefore,the direct CFD comparison of the detailedflow and heat transfer in the backface clearance is difficult.However, the CFD verification for turbine B and turbine C could provide an indication for the analysis of backface clearanceflow in the fol-lowing sections3.2–3.6.The overall performance comparisons for turbine B are given in Table3.In order to evaluate the influence of the backface clear-ance on the overall performance and outletflowfield,the calcula-tion results with backface clearance(CFD)and without backface clearance(CFD NBC)are presented.The calculated massflow rate agrees well with the experimental data,and the calculated isen-tropic efficiency is higher by2.3%in comparison with the meas-ured data.This may be due to some idealizing assumptions for CFD,such as the circumferential average treatment for the stator-rotor connection interface,the adiabatic walls,and the smooth passage profile.However,in the case of the calculations without backface clearance,the calculated efficiency is higher than the ex-perimental data by3.4%.Hence a larger error is introduced by ignoring the backface clearance loss.Comparisons of outlet parameter distribution are shown in Fig.5. The calculation of spanwise efficiency is based on the same constant span bands of the stator inlet and rotor outlet.One can see that the calculated total pressure,total temperature,flow angle,and effi-ciency distributions for the case with backface clearance agree well with the experimental data.However,the calculated data without backface clearance,including the total temperature and efficiency, have larger errors especially at the midspan.Generally,the simulation results with backface clearance agree better with the experimental data for turbine B.And CFX success-fully simulates the influence of backface clearanceflow on the outlet flowfield.Although the working condition for turbine B is different Table3Overall performance comparisons for turbine BExp CFD CFD NBCMassflow,kg/s 2.71 2.67 2.67 Power,kW424.3431.4437.3 Efficiency,%88.590.891.9 Fig.5Comparisons of outlet parameter distribution for turbine B:(a)total pressure,(b)total temperature,(c)flow angle,and(d)efficiencyfrom that for turbine A,the flow mechanisms of the backface clear-ance flow are the same.With the similar mesh generations and com-putational settings,the verification for turbine B could provide a good indication for the analysis focused on turbine A.Figure 6shows the comparisons of heat transfer coefficient (H )on the flat tip for turbine C.Although the calculation result under-predicts the heat transfer in the reattachment region (near the 70%axial distance in Fig.6(b )),the distribution trend of the calculated H contour on the tip agrees well with the experimental result;both the sweet spot [16]of low H and the reattachment region of high H are well captured (Fig.6(a )).Figure 7presents the heat transfer coefficient on the suction side squealer tip.The distribution trend of the calculated H on the SSS tip agrees well with that in the experiment,and the overall H is reduced significantly by using the squealer tip.A more clear comparison is shown in Fig.8.The average heat transfer coeffi-cient on the SSS tip is 44%lower than that on the flat tip,and the calculated H agrees well with the experimental result.Figures 6–8show that CFX has the ability to simulate the heat transfer on blade tips.The sweet spot and the reattachment region found in the experiment are well captured.Meanwhile,the reduc-tion of heat transfer coefficient by using the squealer tip is well predicted.Although the geometry of the tip clearance is different from that of the backface clearance,the flow phenomena in the clearances (separation,reattachment,etc.)are similar.The suc-cessful predictions for the tip clearance could provide a good indi-cation for the heat transfer study on the backface clearance.The following sections 3.2–3.6will focus on the flow and heat transfer analyses of backface clearance flow for turbine A. 3.2Flow Structure in the Backface Clearance.An experi-mental study was performed by Dambach et al.[29],who,for the first time,investigated the detailed velocity and pressure fields in the tip clearance of radial turbines.It was indicated that in the inlet section,the tip casing scraping flow caused by the relative casing motion dominated the tip region and the leakage flow rate was small.In the midsection,the scraping flow was weakened.In the outlet section,the scraping flow rate became very small,and the leakage flow rate increased substantially,which led to a large leakage loss.However,in the case of the flow in the backface clearance,the interaction between the backface casing scraping flow and the leakage flow is different.The streamlines in the backface clearance at different radial locations are shown in Fig.9,and the height to width ratio of the clearance is scaled for better visualization.Radial location (RL )is defined to be 0%at the scallop rim and 100%at the blade tip (Fig.2).At the location of RL ¼50%,the backface leakage flow is very strong.It separates near the pressure side and reattaches on the blade backface.After accelerating in the clearance,the leak-age flow injects into the mainstream and a high leakage flow ve-locity region is found near the suction side.The scraping flow is relatively weak and it is reversed by the strong leakage flow in the clearance.However,at the location of RL ¼10%,the velocity of the leakage flow decreases drastically,the backface casing scrap-ing flow almost prohibits the leakage flow injecting into the main-stream,and a vortex is found in the clearance,which means the intensity of the leakage flow is nearly the same as that of the scraping flow.Finally,at the location of RL ¼5%,the scraping flow dominates over theclearance.Fig.6Comparisons of H on the flat tip for turbine C:(a )H con-tour on the tip and (b )distribution of H along the axialdistanceFig.7Comparison of H on the SSS tip for turbineCFig.8Comparison of the overall H on the flat and SSS tips for turbine CThe radial variations of mass flow in the backface clearance are shown in Fig.10.The relative velocity in the direction of blade rota-tion is considered to be the leakage flow and otherwise to be the scraping flow.The net mass flow is defined to be the leakage mass flow minus the scraping mass flow (m Net ¼m Leakage Àm Scraping ),which can be used to reveal the relative intensity of the leakage and the scraping flows.At the uppermost radial locations,the leakage mass flow is larger than the scraping mass flow.However,in the region of RL <10%,the leakage mass flow decreases rapidly to 0and the scraping mass flow increases drastically.It is shown that in the region of RL >10%,the net mass flow is positive (m Net >0),and the leakage flow is the dominant flow.In the region of RL <10%,the net mass flow is negative (m Net <0),and the scrap-ing flow is dominant.Experimental and numerical studies on the heat transfer of tip clearance showed that the highest heat transfer regions were located where the flow reattached on the blade tip surface [19,28].Meanwhile,high heat transfer coefficients were found on the suc-tion surface of the blade due to the impingement effect of the tip leakage vortex [23].Therefore,the flow reattachment and impingement in the backface clearance of deeply scalloped radial turbines are analyzed as follows.The limiting streamlines on the blade backface surface are pre-sented in Fig.11.There is a reattachment line along the radial direction on the backface surface.The leakage flow is strong in the upper radial region;it separates near the pressure side and finally reattaches on the blade backface surface (Fig.9(a )).The streamlines at the midclearance height plane are presented in Fig.12(see Fig.2for the definition of the plane).There is a recircula-tion zone in the lower radial region due to the interaction of the leakage and the scraping flows in the clearance,and the velocity in this region is very low (Fig.9(b )).The leakage flow injects into the mainstream near the suction side and the flow direction is changed to spanwise because of the obstruction of the scraping flow,which results in a "separation line"along the radial direction near the suction side in Fig.12.This separation line ends in the lower radial region because the leakage flow here is very weak and the whole clearance is dominated by the scraping flow.In this situation,some of the scraping flows impinge onto the blade suc-tion surface,which can also be seen in Fig.9(c ).The schematic flow structure in the backface clearance is shown in Fig.13.The leakage and the scraping flows interact with each other and complex separation,and reattachment and impingement of the flow are presented,which have a strong relationship with the heat transfer of the blade backface.3.3Heat Transfer in the Backface Clearance.The heat transfer coefficient on the blade surfaces (blade backface,pressureFig.9Streamlines in the backface clearance at different radial locations:(a )RL 550%,(b )RL 510%,and (c )RL 55%Fig.10Radial variations of mass flow in the backfaceclearanceFig.11Limiting streamlines on the bladebackfaceFig.12Streamlines at the mid clearance height planeand suction surfaces)is shown in Fig.14,and the pressure andsuction surfaces are unfolded for better visualization.On the backface surface,the strong leakage flow in the upper ra-dial region separates and reattaches on the blade backface (Fig.13),which results in high heat transfer coefficients (reattachment region)in Fig.14.The leakage flow rate decreases against the ra-dial direction and so does the intensity of the flow reattachment (Fig.10).Therefore,the heat transfer coefficient decreases against the radial direction in the reattachment region.Finally,in the lower radial region,the leakage flow rate is very small and the scraping flow is dominant,which results in a recirculation zone (Fig.13).Low heat transfer coefficients are found in this zone due to the low velocity and thick boundary layer.However,high heat transfer coefficients are observed on the blade backface surface near the scallop rim because of the impingement effect of the strong scrap-ing flow (impingement region).On the suction surface,there is a footprint of high heat transfer coefficients due to the impingement effect of the backface clear-ance leakage vortex,and a similar result was also found in an axial turbine [23].The leakage vortex flows towards the blade tip under the influence of the centrifugal force and the spanwise sec-ondary flow [12].The heat transfer coefficient decreases along the streamwise direction because the leakage flow rate decreasesagainst the radial direction.In the lower radial region,an impinge-ment region with high heat transfer coefficients is also found.On the pressure surface,the heat transfer coefficient is rela-tively low except in the upper radial region where the impinge-ment of the strong leakage flow is relatively large.The flow structure in the backface clearance provides a good reference for the analysis of heat transfer mentioned above.Mean-while,the effects of the flow reattachment,recirculation,and impingement on the heat transfer coefficient in the backface clear-ance are well captured.3.4Flow Mechanisms in the Backface Clearance.The leakage flow rate in tip clearance increases along the streamwise direction as mentioned in Ref.[29].Conversely,in the case of the backface clearance,the leakage flow rate decreases along the streamwise direction (Fig.10).To explain the flow structure in the backface clearance,the pressure field is analyzed in this section.The pressure difference across the backface clearance is the driving force of the leakage flow whereas the scraping flow obstructs the leakage flow from injecting into the mainstream (Fig.15).Since the intensity of the scraping flow (0.5q U 2)decreases monotonically against the radial direction,the pressure difference becomes the major factor that governs the flows in the backface clearance.Therefore,pressure fields will be presented to explain the interaction of the leakage and the scraping flows in the backface clearance.The flow near the scallop rim of the deeply scalloped radial tur-bine is shown in Fig.16.The inducer of deeply scalloped radial turbine blades is usually designed to be straight and radial to mini-mize the centrifugal stress,and the relative flow velocity is almost radial.In the inlet section,the inviscid momentum equation in the rotating cylindrical coordinate system is shown in Eq.(1).With the assumption of W h ¼0,the equation can be expressed as Eq.(2).It can be seen that the Coriolis force (-2W R x )is responsible for the pressure difference in the circumferential direction,which results in blade loadings in the inlet section.However,around the region of the scallop rim,the flow direction is forced to change from radial to axial (Fig.16),the radial velocity is reduced rapidly and so does the pressure difference (Eq.(2)).The circumferential velocity W h is still very small because at the scallop rim,the blade is still straight and radial.So the contribution of flow curvature in circumferential direction,the major source of blade loading in axial turbines,to the circumferential pressure difference is very small here.Moreover,when the radial turbine rotor is scalloped,the unloading effect of backface clearance further reduces the pressure difference.So we can see that the rapid decrease of pres-sure difference in the lower radial region of the backface clear-ance is due to the sudden reduction of radial velocity and Coriolis force (Eq.(2)).À1q R @Ps @h ¼2W R x þW R @W h @R þx þW h R@W h@hþW Z @W h þW R W h(1)Fig.13Schematic of the backface clearanceflowsFig.14H contour on the blade surfaces (the pressure and suction surfaces areunfolded)Fig.15Interaction of the flows in the backface clearance。