Investigation of Seal Cavity Leakage Flow Effect on Multistage Axial Compressor Aerodynamic Performance with a Circumferentially Averaged Method

applied sciences

Article Investigation of Seal Cavity Leakage Flow Effect on Multistage Axial Compressor Aerodynamic Performance with a Circumferentially Averaged Method

Dong Liang 1 , Donghai Jin 1,2,* and Xingmin Gui 1,2

1 Aeroengine Simulation Research Center, School of Energy and Power Engineering, Beihang University, Beijing 100191, China; [email protected] (D.L.); [email protected] (X.G.) 2 Jiangxi Research Institute, Beihang University, Nanchang 330096, China * Correspondence: [email protected]; Tel.: +86-010-8231-6870

Featured Application: This method can be used to quickly analyze the seal cavity leakage effect during compressor design as well as performance degradation caused by labyrinth wear in multistage axial-flow compressors. After being verified in turbine, it can be further used to consider the influence of seal cavity leakage flow in whole aero-engine simulation.

Abstract: The seal cavity leakage flow has a considerable impact on the performance of the aeroengine, especially on the multistage compressor. Thus, a quasi-three-dimensional simulation program named CAM is developed basing on circumferentially averaged throughflow method. The program enables a rapid diagnosis for the performance degradation of multistage compressor caused by labyrinth wear. The coupling flow field between the seal cavity leakage flow and the main flow field at the root of the shrouded stator of a high-loading three-stage compressor with inlet guide vanes (IGV) was simulated Citation: Liang, D.; Jin, D.; Gui, X. by CAM and the results indicate that seal cavity leakage flow has a significant impact on the overall Investigation of Seal Cavity Leakage performance of the compressor. That is, for a 1% increase in the seal-tooth clearance-to-span ratio, the Flow Effect on Multistage Axial decrease in total pressure ratio was 2.6%, and the reduction in efficiency was 0.6%. Stage performance Compressor Aerodynamic Performance with a Circumferentially shows that the seal cavity leakage flow reduces the pressurization capacity of the current stator and Averaged Method. Appl. Sci. 2021, 11, the work capacity of the downstream rotor, but has little effect on the upstream blade row. Spanwise 3937. https://doi.org/10.3390/ distribution of blade element performance shows that the leakage flow leads to an increased flow app11093937 blockage near the hub, resulting in spanwise migration. The incidence of the stator and rear rotor then change through the entire span. The leakage flow leads to the flow blockage and migration and hence Academic Editor: Kambiz Vafai changes the incidence angle, which results in the deterioration of compressor performance.

Received: 7 March 2021 Keywords: seal cavity leakage flow; shrouded stator; flow field destruction; deterioration; circumfer- Accepted: 24 April 2021 entially averaged throughflow model Published: 27 April 2021

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in 1. Introduction published maps and institutional affil- iations. Deterioration, induced by operational and environmental effects, leads to a gradual decline of the performance of an aircraft engine [1–3]. Therefore, requirements for rapid full- engine simulations occur. Numerical calculation methods for aeroengines can be divided into three different types: zero-dimensional, two-dimensional/quasi three-dimensional and three-dimensional. The zero-dimensional calculation does not need detailed geometric Copyright: © 2021 by the authors. parameters, has very low requirements in terms of computer resources and is very fast. Licensee MDPI, Basel, Switzerland. It is suitable for scheme evaluation and preliminary design. However, the accuracy of such This article is an open access article distributed under the terms and a calculation depends on the characteristic lines of the existing components and reliable conditions of the Creative Commons experimental data. At the same time, the flow field of the main channel and the secondary Attribution (CC BY) license (https:// air system (SAS) of the engine cannot be simulated, so the flow field information cannot be creativecommons.org/licenses/by/ used in the optimization of specific components. 4.0/).

Appl. Sci. 2021, 11, 3937. https://doi.org/10.3390/app11093937 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3937 2 of 15

Compared with the zero-dimensional simulation, 3D simulation can describe the three- dimensional relationship between the components and the whole flow channel, and can be used to discuss the complex physical phenomena and flow field structure inside the components. However, in order to obtain such abundant flow field details, a large amount of computer resources are required. In the current literature, there are a few published studies on 3D high fidelity simulation calculation of an aeroengine. The center for turbulence simulation (CITS) of Stanford University has carried out a series of research aimed at developing integrated high fidelity aeroengine simulation technology and has been able to achieve preliminary simulation of the whole aircraft engine [4,5]. Medic et al. [6] completed the simulation of a 20 degrees sector of a gas turbine engine including the fan, compressors, turbines, the exit nozzle and the combustor. The simulation was performed on two different meshes; one was fine, and the other was coarse. For the coarse mesh which contained 14 million cells, the simulation was performed on 700 processors and required around two weeks to converge [7–10]. Three-dimensional simulation of the aeroengine main channel requires a computational speed of more than 1012 per second [11]. Full 3D simulation requires enormous amounts of relatively accurate data such as boundary conditions and initial conditions, which is not easy to obtain. Wang et al. [12] pointed out that even if the current computing capabilities can meet these requirements, considering the large number of resources required for full 3D calculations, it is still difficult to use the full 3D simulation as a tool for conventional research in Aeroengine Design in the short term. In contrast with 0D and 3D simulation, quasi three-dimensional simulation exhibits unique advantages. First of all, quasi-3D simulation can provide more abundant S2 flow field information. Secondly, quasi-3D simulation is easy to implement due to the small amount of calculation. Finally, for specific cases, the quasi-3D simulation can use empirical or semi-empirical models, such as loss and blockage models obtained from the experimental data, to modify the flow field results, so as to avoid the amplification of the flow field calculation error caused by the limitations of the turbulence model. Last but not least, the simulations of full-engine in the literature mentioned above have not included the SAS, which is known to have a considerable impact on the engine performance [13]. As a part of the SAS, seal cavity leakage flow occurs in the seal cavity at shrouded stator root (in compressors and turbines) or shrouded rotor tip (usually in turbines). Figure1 shows the seal cavity leakage in a compressor with shrouded stator. Although the flow exchange between the main channel and the cavity below the axial gap is often suppressed by engineering measures such as a labyrinth seal, it is difficult to fully eliminate the leakage from the axial gap. In recent years, more and more studies have shown that seal cavity leakage flow will obviously deteriorate the performance of axial compressor and turbine in aeroengines, and such influence cannot be ignored. Usually, for every 1% increase in seal-tooth clearance-to-span ratio, the decrease in pressure rise is 3%, and the reduction in efficiency is 1% [14,15]. As a reference, a usual relationship between efficiency deterioration and blade tip clearance in an axial compressor is that the compressor efficiency decreases by 1.5% when the rotor tip clearance increases by 1% with respect to the span [15–18]. For a cantilevered stator, the decrease ranges from 1.0% to 2.0% [15,17,19]. Mahmood and Turner [20] investigated the effect of seal cavity leakage flow on 1.5 stages of a 10 stage axial flow compressor. Numerical results indicate 0.86% efficiency reduction for a seal clearance of 1.3% span. Kato et al. [21] have performed a full three-dimensional unsteady calculation for a high-speed six-stage advanced axial-flow compressor. The calculation results with a sealed cavity under the stator blade reduce the efficiency by 1.7% compared with the results without a cavity. Similar to compressors, losses caused by leakage through the shrouded cavity contribute significantly to the overall losses of turbines [22]. Rosic et al. [23] compared experimental data and numerical simulations in a three-stage turbine and pointed out that calculation of the leakage flows and cavities is necessary in order to get good agreement between calculation and measurement. The flow in a two-stage LP-turbine at relatively low Reynolds numbers is simulated by Kuerner et al. [24] with an in-house 3D code. The overall isentropic efficiency derived from computations with Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16

Appl. Sci. 2021, 11, 3937 3 of 15 cavities is necessary in order to get good agreement between calculation and measurement. The flow in a two-stage LP-turbine at relatively low Reynolds numbers is simulated by Kuerner et al. [24] with an in-house 3D code. The overall isentropic efficiency derived fromcavities computations lies within 0.2% with of cavities the experimental lies within result, 0.2% and of the that experimental is 1.3% closer thanresult, computations and that is 1.3%without closer cavities. than computations without cavities.

FigureFigure 1. 1. TheThe seal seal cavity cavity leakage leakage in in a a compressor compressor with with shrouded shrouded stator.

ConsideringConsidering the the advantages advantages of ofthe the two-dime two-dimensionalnsional method method in the in simulation the simulation of full- of aeroenginefull-aeroengine and the and non-negligible the non-negligible influence influence of the ofseal the cavity seal leakage cavity leakage flow on flow the perfor- on the manceperformance of turbomachinery, of turbomachinery, we have we reason have reason to believe to believe that a that two-dimensional a two-dimensional throughflow through- programflow program considering considering seal cavity seal cavityleakage leakage flow can flow undoubtedly can undoubtedly assist in assist the inperformance the perfor- predictionmance prediction of an aeroengine. of an aeroengine. Adopting Adopting the model the of Morns model and of Morns Hoare and [25] Hoareand Roberts [25] and et al.Roberts [26], Banjac et al. [and26], BanjacPetrovic and [27] Petrovic introduced [27] the introduced mixing loss, the mixingthe secondary loss, the flow secondary loss and flow the changeloss and of the the change flow angle of the caused flow angleby seal caused cavity byleakage seal cavity flow into leakage their flow throughflow into their program through- basedflow programon a stream based function on a stream approach. function Ricci approach. et al. [28] Ricci introduced et al. [28 the] introduced effect of seal the cavity effect leakageof seal cavityflow into leakage their flow through into their program flow through based programon axisymmetric based on Euler axisymmetric equations in Euler the formequations of source in the terms. form Their of source program terms. can Their reflect program the impact can of reflect seal cavity the impact leakage of flow seal on cavity the mainstream,leakage flow but on thecannot mainstream, directly reflect but cannot the fl directlyow field reflect inside the the flow labyrinth field inside sealed the cavity. labyrinth sealedOur cavity. goal is to develop a two-dimensional steady-state simulation software for aeroengineOur goal design is to and develop quick a analysis two-dimensional using a circumferentially steady-state simulation averaged software method. for While aero- theengine coupled design simulation and quick of analysisthe main using flow aand circumferentially the leakage flow averaged can be conducted method. While for both the compressorcoupled simulation and turbine of thestages, main this flow paper and focuses the leakage on the compressor flow can be side. conducted In our previous for both workcompressor [29], the and IGV turbine and the stages, first stage this paper of a multistage focuses on axial-flow the compressor compressor side. In is ourthe previousresearch object,work [ 29on], which the IGV the and grid the independence first stage of a has multistage been verified axial-flow and compressor the influence is the of researchthe seal cavityobject, leakage on which flow the on grid the independence main flow and has its beenmechanism verified has and been the influenceinvestigated. of the In sealthis paper,cavity the leakage research flow object on the is the main IGV flow plus and the itsfirst mechanism three stages has of beenthe same investigated. multistage In axial- this flow.paper, The the work research of this object article is can the IGVbe divided plus the in three first three parts. stages First, to of investigate the same multistage the influ- axial-flow. The work of this article can be divided in three parts. First, to investigate the ence of seal cavity leakage flow on the multistage compressor performance. Second, to influence of seal cavity leakage flow on the multistage compressor performance. Second, investigate the influence of seal cavity leakage flow on a certain stage or a certain blade to investigate the influence of seal cavity leakage flow on a certain stage or a certain blade row. Third, to investigate the propagation of this influence in multistage compressor. row. Third, to investigate the propagation of this influence in multistage compressor.

2.2. Numerical Numerical Model Model and and Research Research Object Object TheThe governing governing equations equations of of CAM CAM are are the circumferentially averaged Navier-Stokes equations.equations. Spalart-Allmaras (S-A)(S-A) oneone equation equation model model is is adopted adopted for for turbulence turbulence modelling. modelling.Since Since the Machthe Mach number number is below is below 0.3 0.3 in in the th shroudede shrouded cavity cavity and and above above 0.30.3 inin the main passage,passage, the the preconditioning preconditioning te techniquechnique proposed proposed by by Merkle Merkle and and Choi Choi [30] [30] is is applied to avoidavoid the the stiffness stiffness of of compressible compressible equations equations at low low speed. speed. A A time-marching time-marching finite finite volume volume methodmethod is is chosen chosen to to solve equations. The The validation validation of of CAM CAM by by NASA NASA Rotor Rotor 67 67 and a high-loadedhigh-loaded low low speed speed fan fan TA36 TA36 can can be be found found in in references references [31,32]. [31,32]. More More details details for for CAM CAM cancan also also be be reviewed reviewed in in references references [33,34]. [33,34]. InIn the the calculation calculation of of labyrinth labyrinth leakage leakage flow flow,, the the accuracy accuracy of of CAM CAM is is validated validated by by the the datadata of of Prasad [35]. [35]. Prasad et al. tested the straight-through labyrinth shown in Figure 2,, andand obtained the flowflow characteristicscharacteristics of of the the labyrinth labyrinth under under 12 12 working working conditions conditions (pressure (pres- ratio from 1.0031 to 1.8974). See reference [35] for details of test pieces and test equipment. For the case that the clearance between the labyrinth teeth Cl = 0.36 mm, they also calculated with Fluent. The calculation results of CAM are compared with their experimental results and Fluent calculation results. Appl.Appl. Sci. Sci. 2021 2021, 11, 11, x, xFOR FOR PEER PEER REVIEW REVIEW 4 4of of 16 16 Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 16

suresure ratio ratio from from 1.0031 1.0031 to to 1.8974). 1.8974). See See reference reference [35] [35] for for details details of of test test pieces pieces and and test test equip- equip- sure ratio from 1.0031 to 1.8974). See reference [35] for details of test pieces and test equipment.ment. For For the the case case that that the the clearance clearance between between the the labyrinth labyrinth teeth teeth Cl Cl = =0.36mm, 0.36mm, they they also also ment. For the case that the clearance between the labyrinth teeth Cl = 0.36mm, they also Appl. Sci. 2021, 11, 3937 calculatedcalculated with with Fluent. Fluent. The The calculation calculation resu resultslts of of CAM CAM are are compared compared with with their their experi- experi-4 of 15 calculated with Fluent. The calculation results of CAM are compared with their experi- mentalmental results results and and Fluent Fluent calculation calculation results. results. mental results and Fluent calculation results.

FigureFigure 2. 2. Geometry Geometry of of labyrinth. labyrinth. FigureFigure 2.2. GeometryGeometry ofof labyrinth.labyrinth. TheThe computational computational mesh mesh of of CAM CAM is is shown shown in in Figure Figure 3 .3 The. The number number of of grid grid points points is is TheThe computationalcomputational meshmesh* of*of CAMCAM isis shownshown in *in* Figure Figure3 3.. TheThe number number of of grid grid points points is is 5123.5123. The The total total temperature temperature T 0T∗, total, total pressure pressure ∗pp0 and and flow flow direction direction (horizontal (horizontal intake) intake) 5123. The total temperature T0 0,* total pressure p0 and*0 ﬂow direction (horizontal intake) are 5123. The total temperature T0 , total pressure p 0 and flow direction (horizontal intake) aregivenare given given at theat at the inlet.the inlet. inlet. The The The static static static pressure pressure pressurepb is p givenpb is is given atgiven the at outlet.at the the outlet. outlet. Adiabatic Adiabatic Adiabatic no-slip no-slip no-slip condition con- con- are given at the inlet. The static pressure p b is given at the outlet. Adiabatic no-slip con- ditionisdition applied is is applied applied at wall at surface.at wall wall surface. surface. The relationship The The relationship relationship betweenb between between mass ﬂow mass mass parameter flow flow parameter parameter and inlet/outlet and and inin- pressuredition is ratioapplied is shown at wall in surface. Figure 4The. It canrelationship be seen that between the calculated mass flow results parameter of CAM and are inlet/outletlet/outlet pressure pressure ratio ratio is is shown shown in in Figure Figure 4 .4 .It It can can be be seen seen that that th the ecalculated calculated results results of of inlet/outlet good agreement pressure ratio with is the shown experimental in Figure results, 4. It can and be theseen errors that th aree basicallycalculated within results the of CAMCAM are are in in good good agreement agreement with with the the experimental experimental results, results, and and the the errors errors are are basically basically rangeCAM ofare experimental in good agreement uncertainty with (4.00~4.90%). the experimental When results, the inlet/outlet and the errors pressure are ratio basically is in withinwithin the the range range of of experimental experimental uncertainty uncertainty (4.00~4.90%). (4.00~4.90%). When When the the inlet/outlet inlet/outlet pressure pressure thewithin range the of range 1.48 toof 1.88,experimental the calculation uncertainty accuracy (4.00~4.90%). of CAM is When higher the than inlet/outlet that of the pressure Fluent ratioratio is is in in the the range range of of 1.48 1.48 to to 1.88, 1.88, the the calculation calculation accuracy accuracy of of CAM CAM is is higher higher than than that that of of calculationratio is in the performed range of by1.48 Prasad to 1.88, et the al. calculation accuracy of CAM is higher than that of thethe Fluent Fluent calculation calculation performed performed by by Prasad Prasad et et al. al. the Fluent calculation performed by Prasad et al.

FigureFigureFigure 3. 3. 3. ComputationalComputational Computational mesh mesh mesh of of of CAM. CAM. CAM. Figure 3. Computational mesh of CAM.

FigureFigureFigure 4. 4.4. RelationshipRelationship Relationship between betweenbetween mass massmass flow flowflow parame parameterparameterter andand and inlet/outletinlet/outlet inlet/outlet pressure pressure ratio. ratio.ratio. Figure 4. Relationship between mass flow parameter and inlet/outlet pressure ratio. The throughflow program CAM has been applied to the first three stages of a high- loading transonic six-stage compressor with inlet guide vanes (IGV). The flow path with sealing cavity is shown in Figure5, where the geometric structure of the channel, cavity and labyrinth is simplified to some extent. Table1 provide the values of specific blade parameters. In Figure6, the computational mesh is shown. The channel and the mainstream Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16

The throughflow program CAM has been applied to the first three stages of a high- Appl. Sci. 2021, 11, 3937loading transonic six-stage compressor with inlet guide vanes (IGV). The flow path with 5 of 15 sealing cavity is shown in Figure 5, where the geometric structure of the channel, cavity and labyrinth is simplified to some extent. Table 1 provide the values of specific blade parameters. In Figure 6, the computational mesh is shown. The channel and the mainstream flow path grid are matchedflow path at grid the junction are matched interface. at the In junction order to interface. study the In influence order to of study seal leakage the influence of seal flow on the compressorleakage flow as ona whole the compressor and on each as row a whole of blades, and on six each configurations row of blades, were six set configurations up to adjust thewere leakage set up flow to adjust by changing the leakage the si flowze of by the changing tip clearance the size (c) of the tiplabyrinth. clearance (c) of the The setting oflabyrinth. each configuration The setting is described of each configuration in Table 2. When is described the tip clearance in Table2 .is When 0, the theseal tip clearance cavity of the iscorresponding 0, the seal cavity stage of is the removed, corresponding and only stage the ismainstream removed, andflow only path the grid mainstream is flow retained. Thepath silhouettes grid is for retained. different The co silhouettesnfigurations for are different shown in configurations Figure 7. are shown in Figure7.

Figure 5. Flow path. Figure 5. Flow path.

Table 1. Main Tableblading 1. parametersMain blading of parametersthe compressor. of the compressor.

ParametersParameters S0 S0R1 R1S1 S1R2 R2S2 R3 S2 S3 R3 S3 No. of AirfoilsNo. of Airfoils 32 3228 2850 5038 3868 50 68 82 50 82 Chord/mmChord/mm 51.65 51.65103.26 103.2646.27 46.2768.51 33.37 68.51 50.42 33.37 28.38 50.42 28.38 Solidity Solidity 0.86 0.861.74 1.741.35 1.351.49 1.27 1.49 1.39 1.27 1.27 1.39 1.27 Aspect ratioAspect ratio 3.65 3.651.67 1.672.91 2.911.68 2.78 1.68 1.59 2.78 2.39 1.59 2.39 Inlet metal angle/(◦) 0.47 −51.22 44.76 −54.12 44.37 −55.21 45.89 Appl. Sci. 2021, 11, x FOR PEERInlet REVIEW metal angle/(°) 0.47 −51.22 44.76 −54.12 44.37 −55.21 45.89 6 of 16 Outlet metal angle/(◦) 8.16 −40.49 9.60 −42.96 11.51 −43.28 14.21 Outlet metal angle/(°) 8.16 −40.49 9.60 −42.96 11.51 −43.28 14.21

Figure 6. The computational mesh of compressor with seal cavity leakage.

Table 2. Clearance values for the different configurations.

Configuration 1 2 3 4 5 6 c1/mm 0 0.4 0.5 0.5 0.5 0.5 c2/mm 0 0 0 0.5 0.75 1 c1/h1 × 100 0 0.30 0.37 0.37 0.37 0.37 c2/h2 × 100 0 0 0 0.54 0.81 1.08

(a) configuration 1

(b) configuration 2–3

The boundary conditions are as follows: for the main flow of the compressor, the total temperature, total pressure and flow direction are given at the inlet, i.e., standard atmos- pheric conditions and horizontal intake. For different working conditions, given the corresponding static pressure at the outlet hub, the static pressure distribution along the spanwise direction at the outlet is obtained by using the simplified radial equilibrium equation. The numerical simulation was carried out at the design speed, i.e., 12302 rpm. Table 3 shows the number of grid points of configuration 4, and the convergence time is about 1.5 h. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 16

Appl. Sci. 2021, 11, 3937 6 of 15

Figure 6. The computational mesh of compressor with seal cavity leakage.

Table 2. Clearance values for the different conﬁgurations. Table 2. Clearance values for the different configurations.

ConfigurationConﬁguration 1 12 23 34 45 56 6 c1/mm c1/mm 0 00.4 0.40.5 0.50.5 0.50.5 0.50.5 0.5 c2/mm c2/mm0 00 00 00.5 0.50.75 0.751 1 c1/h1 × 100 0 0.30 0.37 0.37 0.37 0.37 c1/h1 × c2/h2100 × 100 0 00.30 00.37 00.37 0.540.37 0.810.37 1.08 c2/h2 × 100 0 0 0 0.54 0.81 1.08

(a) configuration 1

(b) configuration 2–3

Figure 7. SilhouettesFigure 7. Silhouettesfor different for configurations. different conﬁgurations.

The boundaryThe conditions boundary conditionsare as follows: are for as the follows: main flow for the of the main compressor, flow of the the compressor, total the temperature,total total temperature, pressure and total flow pressure direction and are flow given direction at the inlet, are given i.e., standard at the inlet, atmos- i.e., standard pheric conditionsatmospheric and horizontal conditions intake. and horizontal For different intake. working For differentconditions, working given conditions,the cor- given respondingthe static corresponding pressure at static the outlet pressure hub, at thethe outletstatic hub,pressure the staticdistribution pressure along distribution the along spanwise thedirection spanwise at the direction outlet atis theobtained outlet by is obtained using the by simplified using the simplifiedradial equilibrium radial equilibrium equation. Theequation. numerical The numericalsimulation simulation was carried was out carried at the outdesign at the speed, design i.e., speed, 12302 i.e.,rpm. 12,302 rpm. Table 3 showsTable the3 shows number the of number grid points of grid of conf pointsiguration of configuration 4, and the 4, convergence and the convergence time is time is about 1.5 h.about 1.5 h.

Table 3. Number of grid points of conﬁguration 4.

Position S0 with Inlet R1 S1 R2 S2 R3 S3 with Outlet Cavity 1 Cavity 2 Sum No. of grid points 67 × 49 29 × 49 29 × 49 36 × 49 39 × 49 32 × 49 51 × 49 6975 8753 29,595

Table 3. Number of grid points of configuration 4.

Position S0 with Inlet R1 S1 R2 S2 R3 S3 with Outlet Cavity 1 Cavity 2 Sum No. of grid points 67 × 49 29 × 49 29 × 49 36 × 49 39 × 49 32 × 49 51 × 49 6975 8753 29,595

3. Results and Discussion 3.1. Compressor Performance Figure 8 shows the flow field at near-stall operation point of configuration 4 of the compressor with seal cavity leakage flow. Figure 8b shows that, forced by pressure difference, the leakage flows into the seal cavity from the stator trailing edge gap, and then flows into the mainstream from the stator leading edge gap after dissipated in vortex form for many times. It can be seen from Figure 8c,d that the labyrinth (the part marked with black box in Figure 8c) is the main sealing part in the cavity flow. When the leakage flows through the labyrinth tip clearance, the flow path contracts, the velocity increases and the static pressure decreases. After passing through the tip clearance, the flow path expands Appl. Sci. 2021, 11, 3937 rapidly. The kinetic energy of the fluid dissipates in the cavity between the teeth, 7during of 15 which the velocity decreases, and the pressure change is not obvious. After that, the fluid enters into the next labyrinth tip clearance, and the above process is repeated. After the leakage flow through the multi-stage labyrinth structure, the pressure is significantly re- multi-stageduced, and labyrinth the sealing structure, process the is pressure completed. is significantly Such a low-energy reduced, andleakage the sealing flow converges process isinto completed. the main Such stream a low-energy at the leading leakage edge flow of converges the stator, into causing the main losses stream by mixing at the leadingwith the edgemain of stream. the stator, As causinga result, losses the flow by mixingfield at with the root the mainof the stream. stator is As affected a result, and the a flow low fieldMach atnumber the root region of the statoris formed. is affected and a low Mach number region is formed.

Figure 8. Flow ﬁeld at near-stall point of conﬁguration 4.

With the increase of labyrinth tip clearance, the overall performance of the compressor decreases. The compressor characteristics of each configuration are shown in Figure9. It can be seen that, on the one hand, the change in labyrinth tip clearance causes flow rate change near the stall point of the compressor. This is due to the different leakage flow rate in the sealing cavity under different configurations, which makes the stall position of the compressor change. Figure 10 shows the streamlines at near-stall point for 3 different configurations. The minimum compressor inlet flow rate of the configuration 2 is close to that of the configuration 1. It is found in Figure 10 that in both configurations 1 and 2, compressor stall occurs at the root of the third stage stator (S3). When c1 increases to 0.5 mm (configurations 3 and 4), the flow rate near the stall point increases significantly, which means the stall margin decreases. At this time, in contrast to what happens in the first two configurations, the location where stall occurs first moves to the S1 root. It can be seen that when the seal cavity leakage flow increases to a certain extent, the flow field structure of S1 root is destroyed, and the compressor stall occurs ahead of time. With the increase of c2, the minimum flow rate increases further, and the stall starting point becomes S2 root (configurations 5 and 6). Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 16

Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 16

Figure 8. Flow field at near-stall point of configuration 4.

Figure 8. Flow field at near-stall point of configuration 4. With the increase of labyrinth tip clearance, the overall performance of the compressor decreases. The compressor characteristics of each configuration are shown in Figure 9. With the increase of labyrinth tip clearance, the overall performance of the compres- It can be seen that, on the one hand, the change in labyrinth tip clearance causes flow rate sor decreases. The compressor characteristics of each configuration are shown in Figure 9. It canchange be seen near that, the stallon the point one ofhand, the compressor.the change in Th labyrinthis is due tip to cleathe differentrance causes leakage flow flowrate rate changein the near sealing the stallcavity point under of the different compressor. configur Thisations, is due which to the differentmakes the leakage stall position flow rate of the in compressorthe sealing cavity change. under Figure different 10 shows configur the streamlinesations, which at near-stallmakes the point stall positionfor 3 different of the con- compressorfigurations. change. The minimum Figure 10 showscompressor the streamlines inlet flow at rate near-stall of the configuration point for 3 different 2 is close con- to that figurations.of the configuration The minimum 1. It compressor is found in Figureinlet flow 10 that rate in of both the configuration configurations 2 is 1 closeand 2,to compres- that of sorthe stallconfiguration occurs at 1.the It isroot found of the in Figurethird stage10 that stator in both (S3). configurations When c1 increases 1 and 2, to compres- 0.5mm (con- sorfigurations stall occurs 3 atand the 4), root the of flow the ratethird near stage the stator stall (S3).point When increases c1 increases significantly, to 0.5mm which (con- means figurationsthe stall margin3 and 4), decreases. the flow rate At thisnear time,the stall in copointntrast increases to what significantly, happens in whichthe first means two con- thefigurations, stall margin the decreases. location Atwhere this time,stall occursin contrast first tomoves what tohappens the S1 inroot. the Itfirst can two be seencon- that figurations,when the theseal location cavity leakage where stall flow occurs increases first tomoves a certain to the extent, S1 root. the It flow can fieldbe seen structure that of Appl. Sci. 2021, 11, 3937 whenS1 root the sealis destroyed, cavity leakage and theflow compressor increases to stall a certain occurs extent, ahead the of flowtime. field With structure the increase of8 of 15of S1c2, root the is minimumdestroyed, flow and therate compressor increases further, stall occurs and theahead stall of startingtime. With point the becomes increase S2of root c2,(configurations the minimum flow 5 and rate 6). incr eases further, and the stall starting point becomes S2 root (configurations 5 and 6).

(a) Overall performance of mass flow-pressure ratio (b) Overall performance of mass flow-efficiency (a) Overall performance of mass flow-pressure ratio (b) Overall performance of mass flow-efficiency Figure 9. Overall performance of the compressor. FigureFigure 9. 9.OverallOverall performance performance of the of thecompressor. compressor.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16

(a) configuration(a) configuration 2 2

(b) configuration 4

FigureFigure 10. 10. StreamlinesStreamlines at atnear-stall near-stall point point for for 3 different 3 different configurations. conﬁgurations.

On the other hand, in a specific flow rate range, the compressor performance gradu- ally decreases with the increase of the clearance. The same flow point of each configuration is selected for further analysis. For configurations 1 to 3, only the size of c1 changes, and the compressor performance of configuration 1 is taken as the benchmark. For configurations 3-6, only the size of c2 changes, and configuration 3 is taken as the benchmark. The variation trend of compressor efficiency attenuation with the tip clearance is obtained, as shown in Figure 11. It can be seen from this figure that for cavity 1, when the tip-to- span ratio (the ratio of tip clearance to stator blade height) increases by 1%, the compressor efficiency decreases by 0.4%; for cavity 2, the efficiency decreases by 0.6%, and as the gap continues to increase, the extent of efficiency decrease has an increasing trend. It is in the same order of magnitude as 1% efficiency decay in reference [15].

(a) Clearance 1 (b) Clearance 2 Figure 11. Efficiency penalty for varying seal-tooth clearance. Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16

(b) configuration 4

Appl. Sci. 2021, 11, 3937 9 of 15 (c) configuration 5 Figure 10. Streamlines at near-stall point for 3 different configurations.

OnOn the the other other hand, hand, in a in specific a specific flow flow rate rate range, range, the thecompressor compressor performance performance gradu- grad- allyually decreases decreases with with the increase the increase of the of cleara the clearance.nce. The same The flow same point flow of point each configura- of each con- tionfiguration is selected is for selected further for analysis. further For analysis. configurations For configurations 1 to 3, only 1 the to 3,size only of c1 the changes, size of c1 andchanges, the compressor and the compressorperformance performance of configurat ofion configuration 1 is taken as 1 the is taken benchmark. as the benchmark. For con- figurationsFor configurations 3-6, only the 3–6 size, only of c2 the changes, size of and c2 changes, configuration and configuration 3 is taken as the 3 is benchmark. taken as the Thebenchmark. variation trend The variation of compressor trend ofefficiency compressor attenuation efficiency with attenuation the tip clearance with the is tip obtained, clearance as isshown obtained, in Figure as shown 11. It in can Figure be seen 11. Itfrom can this be seen figure from that this for figure cavity that 1, when for cavity the tip-to- 1, when spanthe ratio tip-to-span (the ratio ratio of tip (the clearance ratio of tipto stat clearanceor blade to height) stator bladeincreases height) by 1%, increases the compressor by 1%, the compressor efficiency decreases by 0.4%; for cavity 2, the efficiency decreases by 0.6%, and efficiency decreases by 0.4%; for cavity 2, the efficiency decreases by 0.6%, and as the gap as the gap continues to increase, the extent of efficiency decrease has an increasing trend. continues to increase, the extent of efficiency decrease has an increasing trend. It is in the It is in the same order of magnitude as 1% efficiency decay in reference [15]. same order of magnitude as 1% efficiency decay in reference [15].

(a) Clearance 1 (b) Clearance 2

FigureFigure 11. 11. EfficiencyEfﬁciency penalty penalty for for varying varying seal-tooth seal-tooth clearance. clearance.

3.2. Stage Performance In order to explore the influence of flow rate of seal cavity leakage on the performance of all stages of the compressor, the pressure coefficient (ψ) and work coefficient (ψ0) of each stage under the corresponding state were calculated by using the flow field data of selected working points. The definition of ψ and ψ0 are as follows

H∗ − H∗ = out in isen ψ 2 Um

H∗ − H∗ 0 = out in ψ 2 Um

where Um is the tangential speed of the rotor at 50% of the span at the leading edge of the stage. Referring to the analysis method in Figure 11, configurations 1–3 are a group and configuration 1 is considered as a benchmark to study the effect of changes in c1 on the characteristics of each stage. Configurations 3–6 are another group and configuration 3 is considered as a benchmark to study the effect of changes in c2 on the characteristics of each stage. Figures 12 and 13 show the variation of work coefficient attenuation and pressure coefficient attenuation with respect to tip clearance, respectively. From the first stage characteristic attenuation (blue line) of Figures 12b and 13b, it can be seen that the leakage flow in the seal cavity 2 under the second stage stator has little effect on the performance of the first stage compressor. As it can be seen from Figure 12a, the leakage flow caused by the seal cavity 1 has little influence on the work capacity of the first stage (current stage), but has a relatively large impact on the latter two stages, thus reducing its work capacity. From Figure 12b, it can be concluded that seal cavity 2 has little effect on the work capacity of the current stage, almost no effect on the front stage (the first stage), and reduces the work capacity of the later stage (the third stage). On the other hand, as it can be seen from Figure 13, the seal cavity leakage flow has little effect on the pressurization capacity of Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 16

3.2. Stage Performance In order to explore the influence of flow rate of seal cavity leakage on the performance of all stages of the compressor, the pressure coefficient (ψ ) and work coefficient (ψ ' ) of each stage under the corresponding state were calculated by using the flow field data of selected working points. The definition of ψ and ψ ' are as follows

()HH**− ψ = out in isen 2 Um ()HH**− ψ ' = out in U 2 m

Where Um is the tangential speed of the rotor at 50% of the span at the leading edge of the stage. Referring to the analysis method in Figure 11, configurations 1–3 are a group and configuration 1 is considered as a benchmark to study the effect of changes in c1 on the characteristics of each stage. Configurations 3–6 are another group and configuration 3 is considered as a benchmark to study the effect of changes in c2 on the characteristics of each stage. Figures 12 and 13 show the variation of work coefficient attenuation and pressure coefficient attenuation with respect to tip clearance, respectively. From the first stage characteristic attenuation (blue line) of Figures 12b and 13 b, it can be seen that the leakage flow in the seal cavity 2 under the second stage stator has little effect on the performance of the first stage compressor. As it can be seen from Figure 12a, the leakage flow caused by the seal cavity 1 has little influence on the work capacity of the first stage (current stage), but has a relatively large impact on the latter two stages, thus reducing its work capacity. From Figure 12b, it can be concluded that seal cavity 2 has little effect on Appl. Sci. 2021, 11, 3937 10 of 15 the work capacity of the current stage, almost no effect on the front stage (the first stage), and reduces the work capacity of the later stage (the third stage). On the other hand, as it can be seen from Figure 13, the seal cavity leakage flow has little effect on the pressurizationthe front stage capacity (the change of the of cavityfront stage 2 in Figure (the change 13b does of notcavity affect 2 in pressure Figure 13 coefficientb does not of affectthe first pressure stage), coefficient but reduces of the the first pressurization stage), but reduces capacity the of thepressurization current stage capacity and the of nextthe currentone. Considering stage and the that next the one. work Considering capacity isthat mainly the work related capacity to the is rotormainly blades, related and to the the rotorpressurization blades, and capacity the pressurization is affected by capacity both the is rotoraffected and by stator, both thethe followingrotor and conclusionstator, the followingcan be drawn: conclusion Seal cavity can be leakage drawn: flow Seal affects cavity the leakage performance flow affects of the the stator performance row and theof theblade stator row row located and the at downstream,blade row located but the at downstream, impact on the but upstream the impact blade on row the (includingupstream bladethe rotor row blade(including row of the the rotor same blade stage row and of the the more same upstream stage and blade the more row) upstream is minimal. blade This row)conclusion is minimal. will beThis further conclusion confirmed will inbe the further analysis confirmed of the spanwise in the analysis distribution of the ofspanwise the flow distributionparameters of of the flow blade parameters row. of the blade row.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 16

(a) Clearance 1 (b) Clearance 2

FigureFigure 12. 12. StageStage work work coefficient coefﬁcient penalty penalty for for varying varying seal-tooth seal-tooth clearance. clearance.

(a) Clearance 1 (b) Clearance 2

FigureFigure 13. 13. StageStage pressure pressure coefficient coefﬁcient penalty penalty for for varying varying seal-tooth seal-tooth clearance. clearance.

3.3.3.3. Spanwise Spanwise Parameters Parameters InIn order order to to further further analyze analyze the the flow flow mech mechanismanism of of the the change change of of compressor compressor stage stage characteristicscharacteristics caused caused by by seal seal cavity cavity leakage leakage flow, flow, the the distribution distribution of of spanwise spanwise parameters parameters reflectingreflecting the the performance performance of ofblade blade row row under under different different tip clearance tip clearance conditions conditions were werean- alyzedanalyzed and andcompared. compared. Figure Figure 14 shows 14 shows the inlet the axial inlet momentum axial momentum of the second of the secondstage stator stage S2stator (normalized S2 (normalized by the inlet by the axial inlet momentum axial momentum at the middle at themiddle span of span the second of thesecond stage rotor stage whenrotor there when is there no gap is no leakage), gap leakage), incidence incidence angle, angle,diffusion diffusion factor factorand loss and coefficient. loss coefficient. The definitionThe definition of diffusion of diffusion factor factor D and D loss and coefficient loss coefficient ϖ andv and are areas follows as follows w ∆w D = 1 − w2 + Δwu D =−1 w2 +2τwu 1 τ 1 ww112 p**− p ϖ = 12 * − p11p

where w1 is the average velocity at the blade inlet, w 2 is the average velocity at the Δ τ * blade outlet, wwwuuu=-12 is the torsional velocity, is the blade solidity, p1 is * the average total pressure at blade inlet, p 2 is the average total pressure at blade outlet,

and p1 is the average static pressure at blade inlet. In general, the change of cavity 1 clearance has little effect on S2, but with the increase of cavity 2 clearance c2, the inlet axial momentum of S2 blade below 15% span decreases, reflecting the increase of flow blockage; the incidence angle also increases obviously. The difference of incidence angle between c2 maximum clearance and non-clearance case reaches 6 ° at 0.25% span, which also confirms the increase of flow blockage and blade load. The flow blockage forces the fluid to migrate to the middle of the span, resulting in a decrease of the incidence angle of the blade above 15% of the span. The difference in the incidence angle between 20% and 40% relative span is the largest. The leakage flow of the seal cavity with an increase of c2 from 0 to 1 mm reduces the incidence angle by 1.6 °. The decrease of incidence angle leads to the decrease of blade load. In contrast to the other parameters, the change of loss mainly occurs at the root of the stator blade, which is shown as a pronounced bulge in the figure. With the increase of c2, the maximum loss coefficient of stator root increases from 0.2 to 0.55, and the spanwise range of bulge also expands from 5% to 18%. Appl. Sci. 2021, 11, 3937 11 of 15

∗ ∗ p1 − p2 v = ∗ p1 − p1

where w1 is the average velocity at the blade inlet, w2 is the average velocity at the blade ∗ outlet, ∆wu = |w1u − w2u| is the torsional velocity, τ is the blade solidity, p1 is the average Appl. Sci. 2021, 11, x FOR PEER REVIEW ∗ 12 of 16 total pressure at blade inlet, p2 is the average total pressure at blade outlet, and p1 is the average static pressure at blade inlet.

Figure 14.14. Spanwise distribution of blade elementelement performanceperformance forfor StatorStator 2.2.

InFigures general, 15 theand change 16 show of the cavity spanwise 1 clearance distribution has little of effect blade on row S2, butparameters with the of increase rotors ofR2 cavityand R3, 2 clearance respectively. c2, the The inlet diffusion axial momentum factor of R2 of and S2 bladethe loss below coefficient 15% span of both decreases, rotor reflectingblade rows the are increase not shown of flowsince blockage;they change the little. incidence In addition, angle in also order increases to make obviously. the con- Thetrast difference more obvious, of incidence Figure 15 angle only betweenshows the c2 distribution maximum clearance of R2 below and 30% non-clearance span. It should case reachesbe pointed 6◦ atout 0.25% that since span, the which rotation also direction confirms of the the increase rotor is defined of flow as blockage the positive and bladedirec- load.tion of The the flowcircumferential blockage forces angle the in fluidthe calculation to migrate process, to the middle it can be of seen the span, from resultingthe velocity in atriangle decrease of the of the elementary incidence flow angle that of when the blade the axial above momentum 15% of the of span. the rotor The inlet difference is small, in the incidence angle is between numerically 20% and negative, 40% relative which is span different is the from largest. that The of the leakage stator. flow From of ◦ theFigure seal 15 cavity it can be with seen an that increase the inlet of c2 axial from momentum 0 to 1 mm and reduces incidence the incidence angle of R2 angle is basically by 1.6 . Theunchanged decrease with of incidence the change angle of c2, leads that to is, the the decrease leakage of flow blade from load. the In contrastseal cavity to thehasother little parameters,effect on the the performance change of loss of upstream mainly occurs blade at row. the rootWith of the the increase stator blade, of c1, which the inlet is shown axial asmomentum a pronounced of R2 bulge root in (below the figure. 8% span) With thedecreases, increase and of c2, the the incidence maximum angle loss coefficientincreases, ofwhich stator reflects root increases that the from flow 0.2 is toblocked. 0.55, and The the blockage spanwise causes range ofthe bulge flow alsoto migrate expands to from the 5%middle to 18%. of the span. In consequence, the inlet axial momentum above 10% span increases, and theFigures incidence 15 and angle 16 show decreases. the spanwise It can be distribution seen that ofthe blade change row of parameters leakage flow of rotorsin the R2S1 androot R3,also respectively. affects the performance The diffusion of factor downstream of R2 and R2 theblade loss row. coefficient Similar of to both Figure rotor 15, Figure blade rows16 shows arenot that, shown with sincethe increase they change of the little. tip clea In addition,rance c2 of in the order seal to cavity, make the the contrast leakage more flow obvious,also causes Figure root 15 blockage only shows of R3. the The distribution blockage of further R2 below causes 30% the span. flow It shouldto migrate be pointed to the outmiddle that of since the thespan. rotation In most direction spanwise of range, the rotor the is incidence defined as angle the positiveof R3 decreases, direction as of well the circumferentialas the load and anglethe work in the capacity. calculation The process,change itof canload be is seen also from reflected the velocity in the diagram triangle ofof the elementarydiffusion factor. flow that when the axial momentum of the rotor inlet is small, the incidence angle is numerically negative, which is different from that of the stator. From Figure 15 it can be seen that the inlet axial momentum and incidence angle of R2 is basically unchanged with the change of c2, that is, the leakage flow from the seal cavity has little effect on the performance of upstream blade row. With the increase of c1, the inlet axial momentum of R2 root (below 8% span) decreases, and the incidence angle increases, which reflects that the flow is blocked. The blockage causes the flow to migrate to the middle of the span. In consequence, the inlet axial momentum above 10% span increases, and the incidence angle decreases. It can be seen that the change of leakage flow in the S1 root also affects the performance of downstream R2 blade row. Similar to Figure 15, Figure 16 shows that, with the increase of the tip clearance c2 of the seal cavity, the leakage flow also causes root blockage of R3. The blockage further causes the flow to migrate to the middle of the span. In most spanwise range, the incidence angle of R3 decreases, as well as the load and the workFigure capacity. 15. Spanwise The distribution change of loadof blade is also element reflected performance in the diagram for Rotor of2. the diffusion factor. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 16

Figure 14. Spanwise distribution of blade element performance for Stator 2.

Figures 15 and 16 show the spanwise distribution of blade row parameters of rotors R2 and R3, respectively. The diffusion factor of R2 and the loss coefficient of both rotor blade rows are not shown since they change little. In addition, in order to make the contrast more obvious, Figure 15 only shows the distribution of R2 below 30% span. It should be pointed out that since the rotation direction of the rotor is defined as the positive direction of the circumferential angle in the calculation process, it can be seen from the velocity triangle of the elementary flow that when the axial momentum of the rotor inlet is small, the incidence angle is numerically negative, which is different from that of the stator. From Figure 15 it can be seen that the inlet axial momentum and incidence angle of R2 is basically unchanged with the change of c2, that is, the leakage flow from the seal cavity has little effect on the performance of upstream blade row. With the increase of c1, the inlet axial momentum of R2 root (below 8% span) decreases, and the incidence angle increases, which reflects that the flow is blocked. The blockage causes the flow to migrate to the middle of the span. In consequence, the inlet axial momentum above 10% span increases, and the incidence angle decreases. It can be seen that the change of leakage flow in the S1 root also affects the performance of downstream R2 blade row. Similar to Figure 15, Figure 16 shows that, with the increase of the tip clearance c2 of the seal cavity, the leakage flow also causes root blockage of R3. The blockage further causes the flow to migrate to the Appl. Sci. 2021, 11, 3937 middle of the span. In most spanwise range, the incidence angle of R3 decreases, as12 well of 15 as the load and the work capacity. The change of load is also reflected in the diagram of the diffusion factor.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 16 Figure 15. SpanwiseSpanwise distribution distribution of blade element performance for Rotor 2.

Figure 16. Spanwise distribution of blade element performance for Rotor 3.3.

Based on thethe performanceperformance of of R2, R2, S2 S2 and and R3 R3 blade blade rows, rows, it canit can be seenbe seen that that the the influence influ- enceof leakage of leakage flow flow of seal of cavityseal cavity on upstream on upstre bladesam blades in multistage in multistage axial-flow axial-flow compressor compressor can canbe ignored. be ignored. On theOn otherthe other hand, hand, it will it will cause cause flow flow blockage blockage at the at rootthe root of current of current stator stator and anddownstream downstream rotor, rotor, and further and further lead to lead flow to migration flow migration to the middle to the middle span. The span. deterioration The dete- riorationof the root of flow the root field flow makes field the makes loss of the the loss stator of the root stator increase root obviously. increase obviously. The spanwise The spanwisemigration migration makes the makes incidence the angleincidence of the angle stator ofand the thestator downstream and the downstream rotor decrease rotor in decreasemost of thein most spanwise of the range, spanwise resulting range, in result the loading in decreases, the load decreases, so does the so pressurizationdoes the pres- surizationcapacity of capacity the blade of rowthe blade and the row work and capacity the work of capacity the rotor. of These the rotor. results These are results consistent are withconsistent those with in Section those 3.2in .Section 3.2. 4. Conclusions 4. Conclusions In this paper, the circumferentially averaged method is used to calculate the coupling In this paper, the circumferentially averaged method is used to calculate the coupling flow field between the seal cavity leakage flow and the main flow field at the root of flow field between the seal cavity leakage flow and the main flow field at the root of the the shrouded stator in a multistage axial-flow compressor. The influence of the leakage shrouded stator in a multistage axial-flow compressor. The influence of the leakage flow flow of the seal cavity on the compressor performance and flow field details is studied of the seal cavity on the compressor performance and flow field details is studied by by changing the tip clearance of the labyrinth. In addition to the analysis of the effect of changing the tip clearance of the labyrinth. In addition to the analysis of the effect of seal seal leakage flow on flow field and compressor performance, the effect of seal leakage flow leakage flow on flow field and compressor performance, the effect of seal leakage flow on on a particular blade row in multistage environment and its propagation in multistage a particular blade row in multistage environment and its propagation in multistage axial axial flow compressor are also studied through the analysis of stage characteristics. The flowsignificant compressor findings are of also this studied study can through be summarized the analysis as of follows: stage characteristics. The significant findings of this study can be summarized as follows: (1) With the increase of leakage flow, the total pressure ratio and efficiency of the com- (1) With the increase of leakage flow, the total pressure ratio and efficiency of the pressor decrease to some extent. The labyrinth clearance-to-span ratio of cavity 1 compressor decrease to some extent. The labyrinth clearance-to-span ratio of cavity 1 in- increases by 1%, and the compressor efficiency decreases by 0.4%. For cavity 2, the creases by 1%, and the compressor efficiency decreases by 0.4%. For cavity 2, the decrease decrease in efficiency is 0.6% when labyrinth clearance-to-span ratio increases by 1%. in efficiency is 0.6% when labyrinth clearance-to-span ratio increases by 1%. Therefore, in Therefore, in order to predict the performance of the compressor correctly, the effect order to predict the performance of the compressor correctly, the effect of leakage flow in of leakage flow in the seal cavity should not be ignored. the seal cavity should not be ignored. (2) The leakage flow in the seal cavity has little effect on the performance of the upstream blade row, but will deteriorate the flow field and performance of the stator and the downstream rotor where the leakage occurs. Reflected in the stage characteristics, it has no effect on the front stage, but has a little influence on the work capacity of the current stage, while the work capacity of the later stage and the pressurizing capacity of the current stage and the rear stage reduces significantly. This is reflected in the performance of blade row that the total pressure loss at the root of stator increases obviously, and the load of stator and downstream rotor decreases in most of the span ranges. (3) The seal cavity leakage flow causes the blockage of the flow at the root of the blade, which leads to the spanwise migration of the flow. The blockage and spanwise migration further change the incidence angle characteristics of the blade row. As a conse- Appl. Sci. 2021, 11, 3937 13 of 15

(2) The leakage flow in the seal cavity has little effect on the performance of the upstream blade row, but will deteriorate the flow field and performance of the stator and the downstream rotor where the leakage occurs. Reflected in the stage characteristics, it has no effect on the front stage, but has a little influence on the work capacity of the current stage, while the work capacity of the later stage and the pressurizing capacity of the current stage and the rear stage reduces significantly. This is reflected in the performance of blade row that the total pressure loss at the root of stator increases obviously, and the load of stator and downstream rotor decreases in most of the span ranges. (3) The seal cavity leakage flow causes the blockage of the flow at the root of the blade, which leads to the spanwise migration of the flow. The blockage and spanwise migration further change the incidence angle characteristics of the blade row. As a consequence, the incidence angle increases at the hub, which induces an increase of total pressure loss. At midspan, on the other hand, the incidence angle decreases, resulting in a decrease of blade load. Generally, the circumferentially averaged throughflow program CAM used in this paper reasonably reflects the influence of seal cavity leakage flow on the performance and flow field details of a three-stage axial-flow compressor. The leakage flow from the sealed cavity will deteriorate the performance of the stator, and this effect will propagate downstream. This program can be used to quickly analyze the seal cavity leakage effect during compressor design as well as performance degradation caused by labyrinth wear in a multistage axial-flow compressor. The study lays a solid foundation for further considering the influence of seal cavity leakage flow in the calculation of aeroengine.

Author Contributions: Conceptualization, D.L. and X.G.; methodology, D.L.; software, D.L.; validation, D.L.; formal analysis, D.L.; investigation, D.L.; resources, D.L.; data curation, D.L.; writing— original draft preparation, D.L.; writing—review and editing, D.L., D.J. and X.G.; project administra- tion, D.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Science and Technology Major Project (2017-I-0005-0006). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data sharing is not applicable to this article. Acknowledgments: The authors would like to thank Fang-fei Ning for the guidance and kind advice, and the fellow apprentices in the research group for their help. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviations

Nomenclature b throttle clearance length (tooth tip thickness) c labyrinth clearance d diameter of labyrinth tooth tip D diffusion factor h blade height hL labyrinth tooth height H∗ total enthalpy i incidence p∗ total pressure S labyrinth pitch T∗ total temperature Um tangential speed of the rotor at 50% of the span at the leading edge of the stage w relative velocity wu tangential component of relative velocity ∆wu torsional velocity Appl. Sci. 2021, 11, 3937 14 of 15

Greek letters η adiabatic efficiency ψ pressure coefficient ψ0 work coefficient v total pressure loss coefficient τ blade solidity Subscripts in the inlet of stage out the outlet of stage isen isentropic

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