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Design and Optimization of Multiple Circumferential Casing Grooves Distribution Considering Sweep and Lean Variations on the Blade Tip Weimin Song, Yufei Zhang and Haixin Chen * School of Aerospace Engineering, Tsinghua University, Beijing 100084, China; [email protected] (W.S.); [email protected] (Y.Z.) * Correspondence: [email protected]; Tel.: +86-010-6278-9269 Received: 18 August 2018; Accepted: 9 September 2018; Published: 11 September 2018







Abstract: This paper focuses on the design and optimization of the axial distribution of the circumferential groove casing treatment (CGCT). Effects of the axial location of multiple casing grooves on the flow structures are numerically studied. Sweep and lean variations are then introduced to the blade tip, and their influences on the grooves are discussed. The results show that the ability of the CGCT to relieve the blockage varies with the distribution of grooves, and the three-dimensional blading affects the performance of both the blade and the CGCT. Accordingly, a multi-objective optimization combining the CGCT design with the sweep and lean design is conducted. Objectives, including the total pressure ratio and the adiabatic efficiency, are set at the design point; meanwhile, the choking mass flow and the near-stall performance are constrained. The coupling between the CGCT and the blade is improved, which contributes to an optimal design point performance and a sufficient stall margin. The sweep and lean in the tip redistribute the spanwise and chordwise loading, which enhances the ability of the CGCT to improve the blade’s performance. This work shows that the present CGCT-blade integrated optimization is a practical engineering strategy to develop the working capacity and efficiency of a compressor blade while achieving the stall margin extension. Keywords: circumferential groove casing treatment; integrated optimization

sweep and lean;

CGCT-blade

1. Introduction The compressor of modern aero-engines has a high stage loading to fulfill the requirement of a high thrust–weight ratio. A high stage loading increases the potential risks of tip stall in some rotors [1]. A casing treatment is usually adopted to enhance the rotor’s stability when the stall margin of the tip-critical rotor is insufficient. Two types of casing treatment, namely, axial slots and circumferential grooves (see Figure 1), are commonly used. Although the circumferential groove casing treatment (CGCT) typically generates less stall margin improvement, it has a smaller efficiency penalty and greater mechanical integrity than axial slots [2,3]. Consequently, a CGCT is more practical than axial slots if the efficiency cannot be sacrificed. The mechanism of a CGCT for extending the stall margin is relevant to the alteration of the flow structures near the blade tip. Rabe and Hah [4] conducted experimental and numerical investigations on a rotor with three different CGCT configurations. They found that the CGCT could reduce the incidence angle and suppress the flow separation. Müller et al. [5] simulated four CGCT configurations under both design and off-design conditions, and they noticed that the grooves segmented the tip leakage vortex (TLV) and alleviated the blockage, which delayed the spillage of the low-energy fluid from the leading edge of the adjacent blade. Sakuma et al. [6] studied the effects of a single

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segmented the tip leakage vortex (TLV) and alleviated the blockage, which delayed the spillage of the low-energy fluid from the leading edge of the adjacent blade. Sakuma et al. [6] studied the effects of a single circumferential groove with locations on NASA Rotor 37. Theythat concluded circumferential groove with different axialdifferent locationsaxial on NASA Rotor 37. They concluded the thatreduced the CGCT reduced the and tip loading and the momentum of theflow. tip leakage flow. CGCT both the tip both loading the momentum of the tip leakage

(a)

(b)

Figure 1. Illustration axial slotsand andcircumferential circumferentialgrooves groovesinina atop topview: view: (a) (a) axial axial slots; slots; (b) Figure 1. Illustration of of axial slots circumferentialgrooves. grooves. (b) circumferential

TheThe axial location of the circumferential grooves significantly affects the the rotor’s performance. axial location of the circumferential grooves significantly affects rotor’s performance. However, the design guidelines for the CGCT’s axial distribution have not been unified. Houghton and However, the design guidelines for the CGCT’s axial distribution have not been unified. Houghton Dayand [7] examined the effectsthe of aeffects singleof circumferential groove at different locations. They reported Day [7] examined a single circumferential groove axial at different axial locations. They thatreported whereas the located at the 8%located and 50% the axial chord (Cax,tip ) achieved thatgroove whereas the groove at points the 8%ofand 50% tip points oflength the axial chord length tip the (𝐶 maximum stall margin improvement, the groove near the leading edge, near the trailingedge, edge,near ) achieved the maximum stall margin improvement, the groove near the leading , andthe at the 18% point Caxatwere inefficient. carried out byresearch Du et al.carried [8] showed that the trailing edge, of and the 18% point ofThe 𝐶 research were inefficient. The out by Du et al. groove located between the 50% and 60% points of C was the most effective for the stall margin [8] showed that the groove located between the 50% was the most effective ax and 60% points of 𝐶 extension; the extension; groove located at approximately 20% point of Cax was actually for the however, stall margin however, the groove located at approximately 20% harmful point ofto𝐶 thewas rotor’s stability. Choi to [9]the tested a single casing groove at different positions. The groove installed near actually harmful rotor’s stability. Choi [9] tested a single casing groove at different positions. the The leading edge was found to be the most effective for expanding the stable operating range with a groove installed near the leading edge was found to be the most effective for expanding the stable small efficiency loss.with The anumerical study of Mao et numerical al. [10] shows that single groove located within operating range small efficiency loss. The study of aMao et al. [10] shows that a single the groove front 40% point of C had obvious benefits to the efficiency and stall margin. Although the axial located within had obvious benefits to the efficiency and stall ax the front 40% point of 𝐶 location of the CGCT plays a significant in affecting the rotor’s performance stability, there margin. Although the axial locationrole of the CGCT plays a significant role inand affecting the rotor’s is not an acknowledged conclusion onnot thisanissue. Nevertheless, most ofon the previous studies used most a performance and stability, there is acknowledged conclusion this issue. Nevertheless, single circumferential groove to simplify the analysis of the effects of the CGCT’s axial location on a of of the previous studies used a single circumferential groove to simplify the analysis of the effects stall.the Those researchers [4,11,12] used multiple grooves[4,11,12] to seek awho larger stall margin grooves improvement CGCT’s axial location onwho a stall. Those researchers used multiple to seek a andlarger higherstall peak efficiency usually chose adjustpeak the depth andusually width ofchose each to groove fixingand margin improvement and to higher efficiency adjustwhile the depth theirwidth axial of locations. Because thefixing effectstheir of multiple groovesBecause are by nothe means a simple accumulation of by each groove while axial locations. effects of multiple grooves are the no individual effects [13], it of is necessary to fillgrooves’ the gap between the itsingle groove study andgap means agrooves’ simple accumulation the individual effects [13], is necessary to fill the the between practical the application of multiple grooves. In this paper, efforts are made to design and optimize single groove study and the practical application of multiple grooves. In this paper, the efforts axial distribution grooves, whichthe alsoaxial contributes to the of understanding of the CGCT’s are made of tomultiple design and optimize distribution multiple grooves, which also influences on the contributes to stall. the understanding of the CGCT’s influences on the stall. TheThe benefits of aof CGCT are are obtained by altering the the tip flow structures. TheThe flowflow structures nearnear benefits a CGCT obtained by altering tip flow structures. structures the the blade tip are alsoalso affected by the 3D shape of the Researchers [14–18] have found thatthat blade tip are affected by the 3D shape of blade. the blade. Researchers [14–18] have found sweeps andand leans can can suppress the the shockwave-boundary layer interaction, redistribute the the spanwise sweeps leans suppress shockwave-boundary layer interaction, redistribute spanwise loading, reduce the the accumulation of the lowlow energy fluids nearnear the the casing andand impact the the strength of of loading, reduce accumulation of the energy fluids casing impact strength the TLV and the flow. Thus, three-dimensional (3D) blade design is supposed interact to the TLV andsecondary the secondary flow.the Thus, the three-dimensional (3D) blade design is tosupposed withinteract the CGCT Houghton Day [7] and proposed theproposed idea combining thecombining casing treatment withdesign. the CGCT design.and Houghton Day [7] the idea the casing design with thedesign blade design to maximize the effectiveness the casing treatment. Recently, [19] treatment with the blade design to maximizeof the effectiveness of the casingHah treatment. studied the inner of axial slots and suggested that slots casing treatments could designed by Recently, Hahworkings [19] studied the inner workings of axial and suggested thatbe casing treatments optimizing the casing groove shape and the blade loading near the tip. However, the topic of the could be designed by optimizing the casing groove shape and the blade loading near the tip. CGCT-blade or optimization has been rarely studied in the open It is in However,integrated the topic ofdesign the CGCT-blade integrated design or optimization has beenliterature. rarely studied noteworthy only Kim al. [11] and Goinis [20]and have done and some pioneer the open that literature. It is et noteworthy that only and KimNickle et al. [11] Goinis Nickle [20]work have in done thissome field. pioneer Moreover, with the increasing use of 3D blading techniques in modern compressor rotors work in this field. Moreover, with the increasing use of 3D blading techniques in andmodern fans, the CGCT design should consider the interactions between the the grooves and a swept or the compressor rotors and fans, the CGCT design should consider interactions between grooves and a swept or leaned blade. With this motivation, effects of the sweep and lean on the CGCT

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leaned blade. With this motivation, effects of the sweep and lean on the CGCT design are studied in this paper. The CGCT-blade integrated optimization is further performed to achieve an ideal CGCT-blade combination. This paper is organized as follows. First, the numerical methods are described and validated. Second, the flow fields of the baseline blade with different CGCTs are simulated and compared. Third, the sweep and lean are introduced to the blade tip. The performance of the modified blade with the CGCT is predicted, and the influences of the 3D blading on the CGCT design are discussed. Finally, the CGCT-blade integrated optimization is conducted. By this process, an optimal CGCT-blade combination is obtained. The mechanism by which the sweep and lean variations affect the effectiveness of CGCT is revealed. 2. Parameterization of the Blade and CGCT The compressor studied in this paper is the Notre Dame Transonic Axial Compressor (ND-TAC) that has been experimentally tested at the University of Notre Dame. The details of the experimental setup can be found in the papers of Cameron et al. [21] and Kelly et al. [22]. The parameters of the original rotor blade are listed in Table 1. Table 1. Design parameters of the Notre Dame Transonic Axial Compressor (ND-TAC) rotor. Symbol

Value

Symbol

Value

Nc (rpm) Utip (m/s) Solidity Nb

14,686 352 1.21 20

Axial Chord (mm) Rcasing (mm) Rhub (mm) τ (mm)

35.56 228.6 171.45 0.762

At the full rotational speed, the corrected choking mass flow of the rotor is approximately 10.1 kg/s and the design mass flow rate is 9.67 kg/s. In addition, the actual mean tip clearance used in the computational fluid dynamics (CFD) simulation is 0.6858 mm. The ND-TAC rotor is tip-critical and it exhibits a spike-type stall behavior [23]. The application of a CGCT to enhance the stability of the ND-TAC [24,25] has demonstrated the effectiveness of CGCTs in this rotor. However, the variation of the design point performance resulting from CGCT was not discussed, or of concern. This paper focuses on the design and optimization of the axial distribution of a CGCT with consideration of the 3D blading design, which aims at developing a novel CGCT-blade integrated optimization strategy to improve the design point performance while extending the stall margin. In this paper, the sweep and lean are introduced to the blade tip to investigate the effects of 3D blading on the CGCT design. The sweep is defined as the shift of the airfoil section along the local chord line. The upstream shift is called the forward sweep, and the downstream shift is the backward sweep. The movement of the section that is perpendicular to the local chord line is defined as the lean. If the direction of the lean is towards the pressure side, the lean is defined as a positive lean; otherwise, it is a negative lean. The blade is generated by a third-order B-spline interpolation based on fourteen airfoils stacked from the hub to the casing. The sweep and lean are manipulated by two third-order Bezier curves. The values of the sweep and lean are defined by the percentage of the baseline-blade mean chord length L and the baseline-blade mean leading-trailing edge deviation ∆(r ·θ ), respectively. Initially, the leading edge (LE) and corresponding trailing edge (TE) for each baseline 2D airfoil are expressed in cylindrical coordinates (r, θ, z). Then, L and ∆(r ·θ ) are calculated by Equations (1) and (2), respectively: L=

q

(z TE − z LE )2 + [(r ·θ )2TE − (r ·θ )2LE ]

(1)

∆(r ·θ ) = |(r ·θ ) TE − (r ·θ ) LE |

(2)

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of all of the 2D airfoils are averaged to obtain the mean value 4 of for 25 the baseline blade. The above definitions of the sweep and lean adopted in this paper are essentially consistent with theand conventional method angle. that defines sweep and leandefined by the by absolute axial axial displacement the circumferential Still, thethe geometry changes the relative displacement and the intuitive. circumferential angle. Still, the changes by the variation can be more Figure 2a,b illustrate thegeometry distributions of thedefined sweep and leanrelative along variation can beThe more intuitive. Figure 2a,b illustrate the distributions the the sweep and lean along the the blade span. abscissa is the percentage of the mean chord lengthofand mean deviation. Both the sweep lean are regulated by five equally root toBoth tip.the In blade span.and Thethe abscissa is the percentage of the mean distributed chord lengthcontrol and thepoints meanfrom deviation. sweep andonly the lean are regulated bythe fiveblade equally distributed controlcauses pointsthe from to tip. In last this this study, the control point at tip is variable, which 3Droot shape of the 35% of only the blade span topoint be affected by the and lean variations. study, the control at the blade tipsweep is variable, which causes the 3D shape of the last 35% of Energies 2018, 11, x FOR PEER REVIEW 4 of 25 the blade span to be affected by the sweep and lean variations. axial displacement and the circumferential angle. Still, the geometry changes defined by the relative variation can be more intuitive. Figure 2a,b illustrate the distributions of the sweep and lean along the blade span. The abscissa is the percentage of the mean chord length and the mean deviation. Both the sweep and the lean are regulated by five equally distributed control points from root to tip. In this study, only the control point at the blade tip is variable, which causes the 3D shape of the last 35% of the blade span to be affected by the sweep and lean variations.

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(b)

Figure Figure 2. 2. Illustration Illustration of ofthe the parameterization parameterization of of the the sweep sweep and and lean: lean: (a) (a) The The sweep; sweep; (b) (b) the the lean. lean. (a) (b)depth With respect respect to to the the CGCT CGCT design, design, thegroove groovenumber, number, width widthand and depthof ofthe thegrooves groovesare arefixed. fixed. With the Figure 2. Illustration of the parameterization of the sweep and lean: (a) The sweep; (b) the lean. Onlythe theaxial axialcoverage coverage CGCT is variable. The parameterization the CGCT is shown in Only ofof thethe CGCT is variable. The parameterization of theof CGCT is shown in Figure Figure 3. The groove number, along with theand depth andofwidth of each are groove areingiven in the right 3. The groove number, along with the depth width each groove given the right side of With respect to the CGCT design, the groove number, width and depth of the grooves are fixed. sidefigure. of theOnly figure. Thecoverage distance between any two grooves remains equal,to facilitate the the the Thethedistance between anyis variable. two adjacent grooves remains facilitate axial of the CGCT Theadjacent parameterization of the CGCTequal, is shown into Figure 3. Theof groove number, along with the depth and width of each groove are giventhe inthe the side of manufacturing of theCGCT. CGCT.In Inthis this figure, theabbreviation abbreviation “TG” means tip gap. The depth manufacturing the figure, the ofof “TG” means tipright gap. The depth of theisfigure. The distance between anyThe two adjacent grooves remains isis equal, to facilitate the of each groove is 4.04 times gapsize. size. Thewidth widthofof eachgroove groove approximately equal to to its its each groove 4.04 times thethe tiptip gap each approximately equal manufacturing of the CGCT. In this figure, the abbreviation of “TG” means the tip gap. The depth of depth. The Thedimension dimensionof ofthe thecasing casinggrooves groovesrefers refersto tothe theprevious previousresearch research [24,25]. [24,25]. A Afew fewCFD CFDtests, tests, depth. each groove is 4.04 times the tip gap size. The width of each groove is approximately equal to its which are are not not provided in this paper, were also carried out to ensure the rationality of the parameter which provided in this paper, were also carried out to ensure the rationality of the parameter depth. The dimension of the casing grooves refers to the previous research [24,25]. A few CFD tests, whichthat are not provided in thisdepth paper, design were alsoare carried ensure the of the parameter selection. Note Note that the width and and depth design are notout thetointerest interest ofrationality this paper; paper; thus, only the the axial axial selection. the width not the of this thus, only selection. Note that the width and depth design are not the interest of this paper; thus, only the axial coverage of of the the overall overall CGCT coverage CGCT configuration configurationinstalled installedon onthe thecasing casingcan canbe becontrolled controlledby bytwo twovariables variablesk coverage of the overall CGCT configuration installed on the casing can be controlled by two variables s, which correspond to the distance between thethe firstfirst groove andand thethe blade’s leading edge andand the 𝑘and and 𝑠, which correspond to the distance between groove blade’s leading edge 𝑘 and 𝑠, which correspond to the distance between the first groove and the blade’s leading edge and distance between two between adjacent grooves, respectively. The variables kvariables and s𝑘are non-dimensionalized the distance two adjacent grooves, The 𝑘 𝑠 and 𝑠 are nonthe between distance two adjacent grooves, respectively. respectively. The variables and are nonby thechord axial length by the axialdimensionalized chordbylength Cax, . chord dimensionalized the axial length 𝐶 𝐶, , . . tip

Figure 3. Illustration the circumferential groove casing casing treatment (CGCT) designdesign parameters. Figure 3. Illustration of theofcircumferential groove treatment (CGCT) parameters.

Figure 3. Illustration of the circumferential groove casing treatment (CGCT) design parameters.

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3. Numerical Methods An in-house CFD code called NSAWET (Navier-Stokes Analysis based on Window-Embedment Technique) is employed to solve the 3D compressible Reynold averaged Navier-Stokes (RANS) equation. An improved version of the HLL-Riemann solver by restoring the contact surface (HLLC) is selected to compute the convective flux [26]. The third-order monotone upstream-centred scheme for conservation laws (MUSCL) [27] with a Van Albada limiter [28] is adopted to ensure accuracy on non-uniform and skew cells. The k-ω shear stress transport (SST) turbulence model [29] is used to compute the turbulent viscosity. Moreover, the time marching is achieved by the Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) implicit algorithm [30]. When the CGCT is adopted, the flux exchange between the groove-casing interfaces are dealt with using an overlap-area weighted reconstruction method [31]. The accuracy of NSAWET as applied to turbomachinery has been validated by previous studies [24,32,33]. The rotor is simulated in the single passage that is discretized by the multi-block structured mesh. The total pressure, total temperature, and axial flow angles are uniformly specified at the inlet boundary. With a given back pressure at the hub, a simplified radial equilibrium equation is employed to compute the radial distribution of the static pressure on the outlet plane. With a steady RANS, the stall point is approached by increasing the back pressure until a further 50 Pa increment will lead to an abrupt mass flow drop and the divergence of the computation. An automatic proportionintegration-differentiation (PID) conditioner [34] is embedded in the NSAWET code to adjust the back pressure until the mass flow converges to the target value at both the design point and the near-stall point. The grid convergence study is conducted with three sets of meshes, corresponding to the coarse mesh, medium mesh, and fine mesh. The total grid numbers of the three sets of meshes are 73.6 million, 1.45 million, and 2.96 million, respectively. Figure 4 shows the details of the medium grid. The grid of circumferential grooves is shown in red, and the grid in the passage is shown in black. This set of mesh has 65 cells distributed along the whole blade span, and another 25 cells inside the tip gap in the radial direction. The wall unit y+ of the first grid layer that is normal to the wall boundary, is strictly maintained below 1, in accordance with the requirements of the SST turbulence model. Each circumferential groove is discretized by a straight H block that has 40, 36, and 50 cells in the stream-, span- and pith-wise directions, respectively. Figure 5 gives a comparison between the CFD results and the experimental data. The uncertainty in the measured total pressure ratio (Pt0, rat ), the measured total temperature ratio (Tt0, rat ), and the measured mass flow are 0.5%, 0.5%, and 1.0%, respectively. Figure 5a,b show the speed curve predicted by the three sets of meshes. All of the three meshes are capable of achieving a good agreement with the experimental data at the design point. The coarse mesh predicts a higher stall mass flow than the experimental data, due to the very sparse grid distribution in the passage. The speed curve obtained by the medium mesh is very close to the curve obtained by the fine mesh over the entire operating mass flow. The differences in the Pt0, rat at the design point and the near-stall point between these two meshes are less than 0.15%, and this value reduces to 0.04% for the Tt0, rat differences. The results of the medium and fine mesh show that the deviation of Pt0, rat between CFD and experimental data occurs near the surge boundary. The present computation also slightly underestimates Tt0, rat . These numerical errors may be caused by the drawbacks of the turbulence model and the insufficient ability of the steady RANS for precisely capturing the large separation under near-stall conditions. Despite the numerical errors, the whole tendency of the speed curve predicted by the medium mesh and fine mesh coincides with the experimental data from choke to stall. In Figure 5c,d, the spanwise distributions of Pt0, rat and Tt0, rat predicted by the medium mesh at the downstream plane also show an acceptable agreement with the experimental data. To sum up, CFD results obtained by the medium mesh are close to the experimental data at the design point and the near-stall point. Because the optimization objectives are set at the design point, and the goal of the optimization is to gain a relative

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improvement based on the baseline blade, it is suitable to use the medium mesh to undertake CFD Energies 2018, 11, x FOR PEER REVIEW 6 of 25 investigations to achieve a favorable balance between the accuracy and the computational cost.

(a)

(b)

(a)

(b)

(c)

(d)

(c) (d) Figure 4. Structured mesh in the computational domain. (a) The geometry of the baseline rotor blade 4. Structured mesh the computational domain. The of the baseline bladeblade with Figure five grooves installed on the casing;(a) (b) thegeometry side view of the the grid rotor distribution on Figure 4. circumferential Structured mesh in in the computational domain. (a) The geometry of baseline rotor with five circumferential grooves installed on the casing; (b) the side view of the grid distribution on the blade surface; (c) the top viewinstalled of the grid on the the side casing; (d)of the grid distribution near with five circumferential grooves ondistribution the casing; (b) view the grid distribution on the blade surface; (c) the top view of the grid distribution on the casing; (d) the grid distribution near the blade leading edge of(c)the tip. of the grid distribution on the casing; (d) the grid distribution near the surface; theblade top view the leading edge of the blade tip. the leading edge of the blade tip.

(a)

(b)

(a)

(b) Figure 5. Cont.

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(c)

(d)

Figure5.5.Validation Validationofofthe thecomputational computationalresults. results.(a) (a)The Thetotal totalpressure pressureratio ratioversus versusthe themass massflow flow Figure rate; (b) the total temperature ratio versus the mass flow rate; (c) the exit total pressure ratio versus rate; (b) the total temperature ratio versus the mass flow rate; (c) the exit total pressure ratio versus thespanwise spanwiselocation locationatatthe themass massflow flowofof9.344 9.344kg/s; kg/s;(d) (d)the theexit exittotal totaltemperature temperatureratio ratioversus versusthe the the spanwiselocation locationatatthe themass massflow flowofof9.344 9.344kg/s. kg/s. spanwise

4.4.CGCT CGCTApplication ApplicationTests Tests The Theeffects effectsofofthe theCGCT CGCTand andits itsaxial axialcoverage coverageare arefirst firstanalyzed analyzedon onthe thebaseline baselineblade bladethat thatisis installed CGCT configurations. Then, thethe sweep andand leanlean are are introduced to the tip. installedwith withdifferent different CGCT configurations. Then, sweep introduced to blade the blade Influences of 3D blading on the performance of the CGCT are discussed. The mass flow rates of the tip. Influences of 3D blading on the performance of the CGCT are discussed. The mass flow rates of design point and near-stall point arepoint taken are as 9.67 kg/s 8.95 kg/s, respectively. abbreviations the design point and near-stall taken asand 9.67 kg/s and 8.95 kg/s, The respectively. The adopted hereafter for each type of rotor are explained in Table 2. abbreviations adopted hereafter for each type of rotor are explained in Table 2. Table Table2.2.Definitions Definitionsofofthe theindividual individualabbreviations. abbreviations. Name Definition Name Definition SW_*solid solid casing rotor, including SW_0,SW_FS, SW_FS, SW_BS, SW_BS, SW_PL and SW_NL SW_* casing rotor, including SW_0, SW_PL and SW_NL CT_* CGCT configuration, including CT_a, CT_b, CT_c, CT_d and CT_e CT_* CGCT configuration, including CT_a, CT_b, CT_c, CT_d and CT_e 0 baseline ND-TAC rotor 0 FS baseline ND-TAC rotor re-stacked rotor with forward swept tip FS BS re-stacked re-stackedrotor rotorwith with forward backwardswept swept tip tip re-stacked rotor with positive leaned BS PL re-stacked rotor with backward swepttiptip re-stacked rotorwith withpositive negative leaned leaned tip PL NL re-stacked rotor tip NL re-stacked rotor with negative leaned tip 4.1. Effects of the CGCT on the Performance of the Baseline Blade 4.1. Effects of the CGCT on the Performance of the Baseline Blade By increasing the parameter k from 0 to 40% of Cax,tip , five CGCT configurations with different axial coverage are the generated. The parameter s isoffixed 5% CGCT of Cax,tip in all five of thedifferent CGCT By increasing parameter 𝑘 from 0 to 40% 𝐶 , as, five configurations with configurations. 3 lists the axial five as CGCT arethe installed axial coverageTable are generated. The coverages parameterof𝑠 these is fixed 5% configurations of 𝐶 , in allwhich five of CGCT on the baseline blade. to CT_e, the position theCGCT CGCTconfigurations is moved fromwhich the leading edge configurations. Table From 3 listsCT_a the axial coverages of theseof five are installed area to the trailing edge area. on the baseline blade. From CT_a to CT_e, the position of the CGCT is moved from the leading edge area to the trailing edge area. Table 3. Five circumferential groove casing treatment configurations for testing.

Table 3. Five circumferential groove casing treatment configurations for testing. Axial Coverage (% Cax,tip ) Name

Name CT_a CT_b CT_a CT_c CT_b CT_d CT_c CT_e CT_d CT_e

Axial Coverage 0–60%(% 𝑪𝒂𝒙,𝒕𝒊𝒑 ) 10–70% 0–60% 20–80% 10–70% 30–90% 20–80% 40–100% 30–90% 40–100%

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Energies 2018, x curves FOR PEER The speed ofREVIEW the ND-TAC Energies 2018, 11,11, 2401

rotor with and without installing the CGCT are shown in25 8 8ofof 25 Figure 6. At both the design point and the near-stall point, all of the CGCT configurations improve Thetotal speed curvesratio of the rotorefficiency with and(𝜂). without installing the of CGCT both the pressure andND-TAC the adiabatic The stall mass flows all ofare the shown groovedin Figure 6.speed Atare both the than design point and thecasing near-stall all of thethe CGCT improve The curves of the ND-TAC rotor withrotor. andpoint, without installing theconfigurations CGCT shown in casing rotor lower that of the solid Specifically, zoom-in of theare stall point in both the total pressure ratio and the adiabatic efficiency (𝜂). The stall mass flows of all of the grooved Figure 6. At both the design point and the near-stall point, all of the CGCT configurations improve Figure 6a shows that the stall mass flows of CT_a, CT_b and CT_c are very close. When the CGCT is casing rotor are lower than thatthe of the solid rotor. the zoom-in ofdecreases, the stall point both theto total ratio and adiabatic efficiency (η). The stall mass flows of all of the grooved moved the pressure trailing edge area, the ability ofcasing the CGCT toSpecifically, delay a stall dramatically whichin Figure 6a shows that the stall mass flows of CT_a, CT_b and CT_c are very close. When the CGCT casing rotor are lower than that of the solid casing rotor. Specifically, the zoom-in of the stall point inis is reflected by the increased stall mass flows of CT_d and CT_e. moved the trailing edge the ability the CGCT to delay stallvery dramatically decreases, which Figure 6atoshows that the stallarea, mass flows of of CT_a, CT_b and CT_caare close. When the CGCT is is reflected the increased stall flows of CT_d CT_e.a stall dramatically decreases, which moved to theby trailing edge area, themass ability of the CGCTand to delay is reflected by the increased stall mass flows of CT_d and CT_e.

(a)

(b)

Figure 6. Speed curves of the baseline blade installed with five CGCT configurations. (a) The total (a) (b) pressure ratio versus the mass flow rate; (b) the adiabatic efficiency versus the mass flow rate. Figure6.6.Speed Speedcurves curvesofofthe thebaseline baselineblade bladeinstalled installedwith withfive fiveCGCT CGCTconfigurations. configurations.(a) (a)The Thetotal total Figure pressure ratio versus the mass flow rate; (b) the adiabatic efficiency versus the mass flow rate. At the design point,the the spanwise distribution of theefficiency total pressure ratio, as flow well rate. as the adiabatic pressure ratio versus mass flow rate; (b) the adiabatic versus the mass

efficiency at 25% of the tip chord length downstream of the trailing edge are shown in Figure 7. In AtAtthe point, the distribution the asasenhances the thedesign design point, thespanwise spanwise distribution thetotal totalpressure pressure ratio, aswell well theadiabatic adiabatic Figure 7a, the CGCT decreases the total pressureofof ratio above 95% ofratio, the as span but the efficiency at 25% of the tip chord length downstream of the trailing edge are shown in Figure 7. InIn efficiency at 25% of the tip chord length downstream of the trailing edge are shown in Figure 7. working capacity of the rest of the span, which demonstrates that the CGCT redistributes the Figure 7a, the CGCT decreases the total pressure ratio above 95% of the span but enhances the working Figure 7a, the CGCT decreases the total pressure ratio above 95% of the span but enhances the spanwise loading. In Figure 7b, the increase of the efficiency concentrated in the blade tip indicates capacity of capacity the rest ofofthethe span, the CGCT the spanwise loading. working restwhich of thedemonstrates span, which that therotor CGCT redistributes the that the CGCT reduces the aerodynamic loss in the that tipdemonstrates region. Theredistributes baseline installed with CT_e In Figure 7b, the increase of the efficiency concentrated in the blade tip indicates that the CGCT reduces spanwise loading. In Figure 7b, the increase of the efficiency concentrated in the blade tip indicates is found to have the highest total pressure ratio below 80% of the span and the highest efficiency the aerodynamic loss the aerodynamic tip region. The baseline rotor installed with CT_e is found to with haveCT_e the that the CGCT loss in the tip region. The baseline rotor installed above 90% of thereduces span.in the highest total below 80% of the span the 80% highest efficiency above of theefficiency span. is found to pressure have theratio highest total pressure ratioand below of the span and the90% highest above 90% of the span.

(a)

(b)

Figure 7.7. Spanwise Spanwise distribution distribution of of (a) (a) the the total total pressure pressure ratio, ratio, and and (b) (b) the the adiabatic adiabatic efficiency efficiency at at the the Figure (a) (b) design point. design point. Figure 7. Spanwise distribution of (a) the total pressure ratio, and (b) the adiabatic efficiency at the design point.

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The design point performance and the stall mass flow of the ND-TAC rotor with different CGCTs The point performance the stall mass flow of the rotor with different installed aredesign summarized in Table 4. Theand performance improvement and ND-TAC the decrease of the stall mass CGCTs installed are summarized in Table 4. The performance improvement and the decrease of the flow due to the installation of different CGCTs are shown in Figure 8. Although the grooved casing stall0mass flowhas due tomost the installation different CGCTs aredesign shownperformance, in Figure 8. Although grooved rotor + CT_e the significantof improvement in the the stall the mass flow rotoris0obviously + CT_e hashigher the most significant improvement the design the stall mass ofcasing 0 + CT_e than the corresponding stallinmass flows performance, of the other CGCT cases. flow of 0 + CT_e is obviously higher than the corresponding stall mass flows of the other CGCT cases. The greatest difference between the five grooved casing rotors is more than 0.1% for the design point The greatest and difference between thefor five grooved casing rotors is more thanCGCT 0.1% for thethe design point performance, is more than 6% the stall mass flow. In the present tests, optimal performance, and is more than 6% for the stall mass flow. In the present CGCT tests, the optimal performance at the design point and near-stall point are not obtained by the same CGCT configuration. performance atofthe point with and its near-stall point are not obtained by the same CGCT The effectiveness the design CGCT varies axial coverage. configuration. The effectiveness of the CGCT varies with its axial coverage.

(b)

(a)

(c) Figure Variation total pressure ratio, (b) adiabatic the adiabatic efficiency, and themass stallflow mass Figure 8. 8. Variation of of thethe (a) (a) total pressure ratio, (b) the efficiency, and (c) the(c) stall flow when different CGCTs are installed on the baseline blade at the design point. when different CGCTs are installed on the baseline blade at the design point. Table 4. 4. Effects of the CGCT on the design pointpoint performance (Pt0, rat and stall Table Effects of the CGCT on rotor’s the rotor’s design performance (𝑃, η) , 𝜂)the and themass stallflow mass , rate (Mrate s ). (𝑀 ). flow Name Name

Pt0, rat 𝑷𝒕𝟎,𝒓𝒂𝒕

Relative to to SW_0 Relative SW_0

η 𝜼

SW_0 SW_0 00++CT_a CT_a 0 + CT_b 0 + CT_b 0 + CT_c CT_c 0 0+ +CT_d CT_d 00++CT_e

1.43655 1.43655 1.44519 1.44519 1.44596 1.44596 1.44475 1.44475 1.44525 1.44525 1.44685

- 0.602% 0.602% 0.655% 0.655% 0.571% 0.571% 0.606% 0.606% 0.717%

0.83008 0.83008 0.83355 0.83355 0.83511 0.83511 0.83323 0.83323 0.83391 0.83391 0.83474

0 + CT_e

1.44685

0.717%

0.83474

Relative to to SW_0 to SW_0 ) Relative Relative SW_0 Ms𝑴(kg/s (kg/s) Relative to SW_0 - 0.418% 0.418% 0.606% 0.606% 0.379% 0.379% 0.461% 0.461% 0.561%

0.561%

𝒔

8.666 8.666 7.882 7.882 7.907 7.907 7.901 7.901 8.089 8.089 8.403

8.403

- −9.044% −9.044% −8.759% −8.759% −8.820% −8.820% −6.663% −6.663% −3.033%

−3.033%

As indicated by Figure 8, the differences between CT_c and CT_e in the design point performance and the stall mass flow are the most obvious. Thus, the flow fields of 0 + CT_c and 0 + CT_e are extracted for comparison and analysis.

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As indicated by Figure 8, the differences between CT_c and CT_e in the design point performance and the stall mass flow are the most obvious. Thus, the flow fields of 0 + CT_c and 0 + CT_e are extracted for comparison and analysis. In Figure 9, the blockage at 93% of the span formed by the TLV after passing the shockwave is Energies 2018, 11, x FOR PEER REVIEW 10 of 25 characterized by the relative Mach number contours. Common effects of CT_c and CT_e on the flow In Figure 9, theare blockage at 93% of span formed by the afterpoint passing shockwave fields in the tip region alleviations of the blockages at both theTLV design andthe the near-stallispoint. by the relativewith Machthe number contours. effects of CT_c and CT_e of onthe the low flowMach Whencharacterized the mass flow decreases variation of theCommon operating point, the behaviors fields in the tip region are alleviations of blockages at both the design point and the near-stall point. number region in 0 + CT_c are basically unchanged (see Figure 9c,d). By contrast, from the design mass flow decreases with the variation of theinoperating the behaviors of the pointWhen to thethe near-stall point, the low Mach number area 0 + CT_epoint, expands rapidly, and thelow average Mach number region in 0 + CT_c are basically unchanged (see Figure 9c,d). By contrast, from the speed in this blockage region is also reduced (see Figure 9e,f). The rapid growth of the blockage in design point to the near-stall point, the low Mach number area in 0 + CT_e expands rapidly, and the 0 + CT_e implies that CT_e that locates between 40% of Cax,tip and 100% of Cax,tip is less effective at average speed in this blockage region is also reduced (see Figure 9e,f). The rapid growth of the delaying the in stall between 20% of C40% of100% Cax,tip ax,tip blockage 0 + than CT_e CT_c impliesthat thatlocates CT_e that locates between of and 𝐶 , 80% and of. 𝐶However, is lessat the , design point, the blockage in 0 + CT_e is smaller than that in 0 + CT_c (see Figure 9c,e), which effective at delaying the stall than CT_c that locates between 20% of 𝐶 , and 80% of 𝐶 indicates . , that CT_e is more relieving the blockage under a high-mass-flow CT_c is. However, at thehelpful design at point, the blockage in 0 + CT_e is smaller than that in 0condition + CT_c (seethan Figure Therefore, the Mach number plotted in Figure 9 demonstrates 0 +a CT_e has a stronger 9c,e), which indicates that contours CT_e is more helpful at relieving the blockagethat under high-mass-flow condition than CT_c is. Therefore, the Mach number contours plotted in Figure 9 demonstrates that the working ability at the design point, but a smaller stall margin than 0 + CT_c has. In addition, 0 + CT_e has a stronger working ability at the design point, but a smaller stall margin than 0 + CT_c blockage that originates downstream of the shockwave and then extends to the aft-part of the pressure has. In originates downstream the shockwave andloading then extends to the side (PS) is addition, supposedthe toblockage affect thethat blade’s aft-loading. If theofblade’s chordwise is redistributed aft-part of the pressure side (PS) is supposed to affect the blade’s aft-loading. If the blade’s chordwise due to the 3D blading, the effectiveness of CGCT can be changed. loading is redistributed due to the 3D blading, the effectiveness of CGCT can be changed.

(a)

(b)

(c)

(d)

Figure 9. Cont.

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(e)(e)

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(f) (f)

Figure 9. The relative Mach number contours on SW_0 the design point; (b)SW_0 Figure 9. relative Mach number contours onon thethe S1 S1 plane forfor (a) SW_0 atatthe point; (b) Figure 9. The The relative Mach number contours the S1plane plane for(a) (a) SW_0 atdesign the design point; (b) SW_0 at the near-stall point; (c) 0 + CT_c at the design point; (d) 0 + CT_c at the near-stall point, (e) at the near-stall point; (c) 0 + CT_c the design point; (d) 0 + CT_c the near-stall point, (e) 0 +0 CT_e SW_0 at the near-stall point; (c) 0 +atCT_c at the design point; (d) 0 +atCT_c at the near-stall point, (e) 0 + CT_e at the design point, and (f) 0 + CT_e at the near-stall point. the design (f) 0and + CT_e the near-stall point. point. +atCT_e at the point, designand point, (f) 0 +atCT_e at the near-stall

Figure 10 quantitatively compares the mass flow distribution along the span at the near-stall

Figure 10 10 quantitatively quantitatively compares compares the the mass mass flow flow distribution distribution along along the the span span at the the near-stall near-stall Figure condition. It shows that both CT_c and CT_e increase the mass flow near the blade tipatdue to the condition. It shows that both CT_c and CT_e increase the mass flow near the blade tip due to the the condition. It shows that bothThe CT_c and CT_e thethemass near the bladethan tip that dueofto reduction of the blockage. mass flow of 0 +increase CT_c near bladeflow tip region is higher reduction of the the blockage. Themass mass flowofof0 0+ drawn +CT_c CT_c near the blade region is higher than that reduction of blockage. The near the blade tiptip region higher than that of 0 + CT_e, which demonstrates the flow conclusion from Figure 9 that CT_c is is more effective to of 0 + CT_e, which demonstrates the conclusion drawn from Figure 9 that CT_c is more effective to 0 + CT_e, which demonstrates the conclusion drawn from Figure is more effective to alleviate the blockage at a low-mass-flow condition. It explains why 09+that CT_cCT_c presents a larger stall alleviate the blockage at a low-mass-flow condition. It explains why 0 + CT_c presents a larger stall margin 0 and 0at+ aCT_e. Moreover, Figure 10 alsoItshows thatwhy the mass flow near the blade root stall alleviate thethan blockage low-mass-flow condition. explains 0 + CT_c presents a larger is reduced by CGCT, which indicates that CGCT not alters the tipmass flow field, but also changes margin than 00 ++ CT_e. Moreover, Figure 10 shows that the the blade root is margin than 00 and and CT_e. Moreover, Figure 10also alsoonly shows that the massflow flownear near the blade root the flow behaviors at the non-tip region. reduced by CGCT, which indicates that CGCT not only alters the tip flow field, but also changes the is reduced by CGCT, which indicates that CGCT not only alters the tip flow field, but also changes flowflow behaviors at the non-tip region. the behaviors at the non-tip region.

Figure 10. The spanwise mass flow distribution at the near-stall point.

Along the trajectory of the TLV, seven planes perpendicular to the Z-axis are extracted, and the entropy contours on them at the design point are shown in Figure 11. The yellow square in the figure Figure spanwise mass flow distribution at near-stall point. Figure 10. 10. The The spanwise flow distribution at the thethe near-stall point.Apparently, the encloses the cross-section where the coremass of the TLV interacts with shockwave. entropy in this area of 0 + CT_e is less than the corresponding entropy of 0 + CT_c, which indicates Along the trajectory trajectory of of the the TLV, TLV, seven planes perpendicular to the are extracted, and the the Along seven planes in perpendicular the Z-axis Z-axis are extracted, and that thethe strength of the shock-TLV interaction 0 + CT_e is to weaker than that in 0 + CT_c. entropy contourson onthem thematatthe the design point are shown inblack Figure 11. in The yellow in the entropy contours design point areby shown in Figure 11. The yellow squaresquare inarea theoffigure Consequently, the high-entropy region marked the dashed square the low-speed

figure encloses the cross-section theofcore the interacts TLV interacts the shockwave. Apparently, encloses the cross-section wherewhere the core theofTLV with with the shockwave. Apparently, the the entropy in this area of 0 + CT_e is less than the corresponding entropy of 0 + CT_c, which indicates entropy in this area of 0 + CT_e is less than the corresponding entropy of 0 + CT_c, which indicates that the of of thethe shock-TLV interaction in 0 + CT_e that inthan 0 + CT_c. Consequently, that thestrength strength shock-TLV interaction in 0 is + weaker CT_e isthan weaker that in 0 + CT_c. Consequently, the high-entropy region marked by the dashed black square in the low-speed area of

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0 + CT_e is smaller. The variations thedashed entropyblack distributions demonstrate that area at theofdesign point, the high-entropy region marked byofthe square in the low-speed 0 + CT_e is CT_e is more effective at increasing the efficiency than CT_c is. smaller. The variations of the entropy distributions demonstrate that at the design point, CT_e is more 0 + CT_e is smaller. The variations of the entropy distributions demonstrate that at the design point, effective at increasing efficiency than CT_c is. than CT_c is. CT_e is more effectivethe at increasing the efficiency

(a)

(b)

(a)contours on the Z-planes at the design point in (a) 0 (b) Figure 11. The entropy + CT_c and (b) 0 + CT_e. Figure 11. 11. The The entropy entropy contours contours on on the the Z-planes Z-planes at at the the design design point point in in (a) (a) 00 ++ CT_c CT_c and and (b) (b) 00 + + CT_e. Figure CT_e.

The distributions of the relative total pressure (𝑃 , ), non-dimensionalized by the inlet total pressure in the tip clearance, are plotted in Figure 12. Figure 12a shows that the by improvement of The distributions of the the inlet inlet total The distributions of the relative relative total total pressure pressure (P (𝑃t0,, rel ), ), non-dimensionalized non-dimensionalized by the total 𝑃 , induced by clearance, CT_e afterare theplotted 40% point of 𝐶 12. is higher the corresponding improvement , Figure pressure in in Figure 12a than shows that the of Pt0, rel pressure inthe thetip tip clearance, are plotted in Figure 12. Figure 12a shows thatimprovement the improvement of for CT_c at the design point; however, at the near-stall point, Figure 12b shows that CT_cfor is CT_c more induced by CT_e after the 40% point of C is higher than the corresponding improvement 𝑃 , induced by CT_e after the 40% point of 𝐶 , is higher than the corresponding improvement ax, tip effective at increasing 𝑃 , at after 30% point of Figure 𝐶 , 12b thanshows CT_ethat is. The variation of 𝑃 , at in at design point; however, thethe near-stall CT_c isthat more effective forthe CT_c at the design point; however, at the point, near-stall point, Figure 12b shows CT_c is more the tip clearance proves that the effectiveness of different CGCT configurations varies with the increasing pointthe of C30% than of CT_e The Pt0, variation ax, tip rel after the rel in the tip effective atPt0, increasing 𝑃 30% after point 𝐶 is. thanvariation CT_e is.ofThe of clearance 𝑃 , in , , operating conditions. proves that the effectiveness of different CGCT configurations varies with the operating conditions. the tip clearance proves that the effectiveness of different CGCT configurations varies with the operating conditions.

(a)

(b)

(a)of Figure 12. 12. Distribution Distribution of the the non-dimensional non-dimensional relative relative total total pressure pressure in in the the(b) tip gap gap at at (a) (a) the the design design Figure tip point and and (b) (b)the thenear-stall near-stallpoint. point. point Figure 12. Distribution of the non-dimensional relative total pressure in the tip gap at (a) the design point andthe (b) the near-stall point.the Effectiveness of the CGCT 4.2. 4.2. Effects Effects of of the Sweep Sweep and and Lean Lean on on the Effectiveness of the CGCT

As Table 4, in both 4.2. Effects of the in Sweep and on theof of the point CGCT As shown shown in Table 4,Lean in terms terms ofEffectiveness both the the design design point performance performance and and the the stall stall margin, margin, the the performance of CT_b is more satisfactory than the other CGCT configurations. In this section, CT_b is performance of CT_b is more satisfactory than the other CGCT configurations. In this section, CT_b As to shown in Table 4,rotor in terms of named both theFS, design point performance and the of stall margin, the applied four re-stacked blades BS, PL, and NL to test the effects the sweep and is applied to four re-stacked rotor blades named FS, BS, PL, and NL to test the effects of the sweep performance of CT_b is more satisfactory than theinother CGCTare configurations. In this section, CT_b lean. The The sweep andand lean distributions adopted study given ininFigure and lean. sweep lean distributions adopted this in this study are given Figure2.2.The The3D 3Dshapes shapes is applied to four re-stacked rotor blades named FS, BS, PL, and NL to test the effects of the sweep of the four blades are plotted in Figure 13. Compared with the baseline ND-TAC blade, the control and lean. The sweep and lean distributions adopted in this study are given in Figure 2. The 3D shapes of the four blades are plotted in Figure 13. Compared with the baseline ND-TAC blade, the control

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Energies 2018,blades 11, x FOR PEER REVIEW 13 of 25 of the four are plotted in Figure 13. Compared with the baseline ND-TAC blade, the control point of the sweep at 100% span is forward and backward shifted by 12% of the mean chord length to pointthe of blades the sweep at 100% span is forward and backward shifted 12%lean of the meanofchord length form FS and BS, respectively. Moreover, the control point by of the at 100% the span is to form the blades FS and BS, respectively. Moreover, the control point of the lean at 100% of the positively and negatively shifted by 3.5% of the mean leading-trailing edge deviation to producespan the is positively shifted by 3.5% the mean edgeindeviation to produce blades PL andand NL,negatively respectively. Although theofposition of leading-trailing the blade tip airfoil the flow passage is the blades PL and NL, respectively. Although the position of the blade tip airfoil in the flow passage changed in accordance with the sweep and lean, the geometry parameters of the 2D airfoil, including is changed accordance with the sweep lean, theAdditionally, geometry parameters ofposition the 2D of airfoil, the thickness,incamber and chord length, are and unchanged. the relative the including the thickness, camber and chord length, are unchanged. Additionally, the relative position circumferential grooves to the blade tip is fixed, regardless of how the blade is re-stacked. of the circumferential grooves to the blade tip is fixed, regardless of how the blade is re-stacked.

(a)

(b)

(c)

(d)

Figure13. 13.3D 3Dshapes shapesofofthe thefour fourre-stacked re-stacked blades: of the forward swept blade FS; the (b) tip the Figure blades: (a)(a) thethe tiptip of the forward swept blade FS; (b) tip of the backward swept blade BS; (c) the tip of the positive leaned blade PL; (d) the tip of the of the backward swept blade BS; (c) the tip of the positive leaned blade PL; (d) the tip of the negative negative leaned leaned blade NL. blade NL.

The TheCFD CFDresults resultsshow showthat thatwhen whenCT_b CT_bisisinstalled, installed,all allfour fourof ofthe there-stacked re-stackedblades bladesare areable ableto to operate operateatatthe themass massflow flowrate rateofof8.10 8.10kg/s, kg/s,which whichisisapproximately approximately0.60 0.60kg/s kg/s lower lowerthan thanthe thestall stallmass mass flow flowrate rateof ofthe thebaseline baselineblade. blade.Therefore, Therefore,the thestall stallmargin marginof ofthe there-stacked re-stackedblades bladeswith withCT_b CT_binstalled installed isissufficient. sufficient.Consequently, Consequently,we wefocus focuson onthe theperformance performanceofofthese theseblades bladesinstead insteadofofthe thestall stallmargin. margin. Figure Figure14 14shows showsthe theperformance performancevariations variationsof ofre-stacked re-stackedblades bladesat atboth boththe thedesign designpoint pointand and the thenear-stall near-stallpoint pointinduced inducedby byCT_b. CT_b. The The forward forward and and backward backward sweep sweep result result in inaaslight slightdesign design performance leads to to greater greater loss loss in in 𝑃 Pt0,, rat and performance deterioration, deterioration, but but the the backward backward sweep sweep leads and η𝜂 than thanthe the forward (see symbol symbol ∆∆and and∇ ∇ in Figure 14a). installing the swept forward sweep sweep (see in Figure 14a). AfterAfter installing CT_bCT_b on theon swept rotor’srotor’s casing, casing, the performance is improved significantly. The performance improvement the performance is improved significantly. The performance improvement inducedinduced by CT_bby inCT_b blade in FS than is more than the corresponding improvement in theblade baseline and blade BS 14b). (see FSblade is more the corresponding improvement in the baseline 0 andblade blade0 BS (see Figure Figure 14b). In addition, after introducing the sweep to the tip, the improvement of the near-stall In addition, after introducing the sweep to the tip, the improvement of the near-stall performance performance induced byand CT_b andthan BS istwice morethe than twice the corresponding improvement in induced by CT_b in FS BSinisFS more corresponding improvement in blade 0 (see blade (see Figure 14d). The positive-leaned PL higher has a slightly higher design performance Figure0 14d). The positive-leaned blade PL has ablade slightly design performance than blade 0. By than bladethe 0. negative By contrast, negative lean causes a performance deterioration 3 14a). and contrast, lean the causes a performance deterioration (see symbol ◇ and(see ○symbol in Figure #The in Figure 14a). The performance improvement after in installing CT_b in blade less than the performance improvement after installing CT_b blade PL is less than PL theiscorresponding corresponding improvement in blade 0; performance however, the improvement performance improvement NLCT_b induced by improvement in blade 0; however, the of NL inducedofby is more CT_b is more we find inFigure blade 014b,d). (see Figure 14b,d). than what wethan findwhat in blade 0 (see From above quantitative comparisons, it canitbecan concluded that the sweep andsweep lean introduced Fromthe the above quantitative comparisons, be concluded that the and lean to the blade tip affect both the working capacity and the efficiency of the blade. The sweep and negative introduced to the blade tip affect both the working capacity and the efficiency of the blade. The sweep lean the solid casingthe rotor’s the effectiveness of the CGCT is and slightly negativedecrease lean slightly decrease solidperformance. casing rotor’sMoreover, performance. Moreover, the effectiveness also altered by is the sweep and by lean. these tests, the effectiveness of the CGCT at improving solidat of the CGCT also altered theInsweep and lean. In these tests, effectiveness of the the CGCT casing rotor’s magnified by the forward sweepbyand in and the tip. improving theperformance solid casing is rotor’s performance is magnified thenegative forward lean sweep negative lean Figure in the tip. 15 shows the variations of the spanwise total pressure ratio distribution caused by the sweepFigure and lean in the the blade tip. In Figure 15a, whereas forward sweep in the tipcaused reduces 15 shows variations of the spanwise totalthe pressure ratio distribution bythe the working ability sweep reduces the working ability of tip thereduces rest of the sweep and leanofinthe theblade bladetip, tip.the In backward Figure 15a, whereas the forward sweep in the the blade span without changing the the tip backward performance. When CT_bthe is installed the casing, the of total working ability of the blade tip, sweep reduces working on ability of the rest the pressure ratio of the forward swept blade FS is greatly improved from the root to 90% of the span. blade span without changing the tip performance. When CT_b is installed on the casing, the total The performance is even larger the corresponding improvement to the baseline pressure ratio of improvement the forward swept blade FS than is greatly improved from the root to 90% of the span. blade with CT_b installed; however, for the backward swept blade BS, the performance improvement The performance improvement is even larger than the corresponding improvement to the baseline induced by CT_b CT_binstalled; is less than the corresponding improvement in FS theperformance same spanwise position. blade with however, for the backward swept blade BS,atthe improvement induced by CT_b is less than the corresponding improvement in FS at the same spanwise position. In Figure 15b, the negative lean reduces the total pressure ratio along the entire span, whereas the positive lean slightly increases the total pressure ratio in the tip. The differences between the grooved

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In Figure 15b, the negative lean reduces the total pressure ratio along the entire span, whereas the Energies 2018, 11, x FOR PEER REVIEW 14 of 25 positive lean slightly increases the total pressure ratio in the tip. The differences between the grooved leaned leaned blades blades PL PL ++ CT_b CT_b and and NL_CT_b NL_CT_b are are not not as asobvious obvious as asthe thedifferences differencesbetween betweenthe thegrooved grooved swept blades FS + CT_b and BS + CT_b. swept blades FS + CT_b and BS + CT_b.

(a)

(b)

(c)

(d)

Figure14. 14.Performance Performancevariations variationsof ofre-stacked re-stackedblades bladesinduced inducedby byCT_b CT_bat atthe thedesign designpoint pointand andthe the Figure near-stall point: (a) P and 𝜂 at the design point; (b) the improvements in P and 𝜂 of the near-stall point: (a) Pt0, t0, rat t0, rat rat and η of the rat and η at the design point; (b) the improvements in Pt0, groovedcasing casingrotors rotorsrelative relativetotothe therespective respectivesolid solid casing rotors design point; and grooved casing rotors at at thethe design point; (c)(c) Pt0,Prat and η t0, rat 𝜂 the at the near-stall point; (d) improvements the improvements inratPt0, 𝜂 of the grooved rotors relative at near-stall point; (d) the in Pt0, and of the grooved casingcasing rotors relative to the rat ηand to the respective solid rotors casingat rotors at the near-stall respective solid casing the near-stall point. point.

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(a)

(b)

Figure 15. Spanwise distribution of the total pressure ratio at the design point for (a) swept blades and (b) leaned blades.

Figure 16 shows the radial relative velocities 𝑊 at each groove-casing interface. 𝑊 varies with the sweep in the tip (see Figure 16a–c). As shown in Figure 16d, compared with the baseline (a) (b) blade, the forward sweep reduces the aspiration of the front three grooves, and the backward sweep Figure 15.aspiration Spanwisedistribution distribution thetotal total pressure ratio at the design point foraspiration (a) swept blades Figure 15. Spanwise ofoftwo the pressure ratio at the design point forthe (a) swept blades increases the of the front grooves while obviously reducing ofand the last and (b) leaned blades. (b) leaned blades. one. Figure 16e) shows that the injection of fluids from the grooves to the passage is suppressed by the forward sweep in the front four grooves. By contrast, in the backward swept blade BS, the Figure Figure 16 16 shows shows the the radial radial relative relative velocities velocities W 𝑊 atat each each groove-casing groove-casing interface. interface. W 𝑊r varies varies injection of the front four grooves is enhanced. To sumr up, the forward sweep tends to decrease the with with the the sweep sweep in in the the tip tip (see (see Figure Figure16a–c). 16a–c). As As shown shown in in Figure Figure 16d, 16d, compared compared with with the the baseline baseline injection and aspiration of grooves, while the backward sweep shows an opposite trend to the blade, forward sweep the aspiration of three grooves, the sweep blade, the the forward sweepofreduces reduces the aspiration of the the front front grooves, and and the backward backward sweep forward sweep in terms affecting the fluid exchange at thethree groove–casing interfaces. The sweep is increases increases the the aspiration aspiration of of the the front front two two grooves grooves while while obviously obviously reducing reducing the the aspiration aspiration of of the the last last found to shift the spanwise loading as well as the chordwise loading [35]; therefore, the aspiration one. Figure shows that ofof fluids from thethe grooves to thethe passage is suppressed by the one.injection Figure16e) 16e) shows thatthe theinjection injection fluids from grooves passage suppressed and of the grooves in the corresponding area vary with the to changes of the is blade loading.by forward sweep in thein front grooves. By contrast, in the backward swept blade BS, the injection of the forward sweep thefour front four grooves. By contrast, in the backward swept blade BS, the Although the relative position between the circumferential grooves and the tip airfoil is the front four grooves is enhanced. Toenhanced. sum up, the forward sweep tends to decrease the injection and injection of the front four grooves is To sum up, the forward sweep tends to decrease the unchanged when the sweep and lean are introduced, the interactions between the blade and CGCT aspiration of grooves, while the backward shows an sweep opposite trendan to opposite the forward sweep in injection and aspiration of grooves, whilesweep the backward shows trend to the are altered, which impacts the performance of the blade and the effectiveness of the grooves. The terms of affecting fluid at the groove–casing interfaces. The sweep is found to sweep shift the forward sweep inthe terms of exchange affecting the fluid exchange at the groove–casing interfaces. The is effects of the 3D blading on the CGCT suggest that it is promising to achieve better design point spanwise loading as well as the chordwise loading [35]; therefore, the aspiration and injection of the found to shift the spanwise loading as well as the chordwise loading [35]; therefore, the aspiration performance while ensuring an adequate stable operating range by considering 3D blading in the grooves in theof corresponding with the changes of the blade loading.of the blade loading. and injection the grooves inarea the vary corresponding area vary with the changes process of the CGCT design. Although the relative position between the circumferential grooves and the tip airfoil is unchanged when the sweep and lean are introduced, the interactions between the blade and CGCT are altered, which impacts the performance of the blade and the effectiveness of the grooves. The effects of the 3D blading on the CGCT suggest that it is promising to achieve better design point performance while ensuring an adequate stable operating range by considering 3D blading in the process of the CGCT design.

(a)

(b)

(c)

Figure 16. Cont.

(a)

(b)

(c)

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(e)

Figure 16. 16. Comparison thethe casing-groove interfaces (a) (a) 0 +0CT_b; (b) Figure Comparisonof ofthe theradial radialrelative relativevelocity velocityatat casing-groove interfaces + CT_b; FS + CT_b; (c) BS + CT_b; (d) maximal radial velocity at each interface (aspiration); (e) minimal radial (b) FS + CT_b; (c) BS + CT_b; (d) maximal radial velocity at each interface (aspiration); (e) minimal velocity at eachatinterface (injection). radial velocity each interface (injection).

Although the relative position between the circumferential grooves and the tip airfoil is unchanged 5. CGCT-Blade Integrated Optimization when the sweep and lean are introduced, the interactions between the blade and CGCT are altered, Because the CGCT-blade interactions are found to play a significant role in altering the rotor’s which impacts the performance of the blade and the effectiveness of the grooves. The effects of the performance and stability, we undertake an integrated optimization of the CGCT-blade combination. 3D blading on the CGCT suggest that it is promising to achieve better design point performance The goal is to verify the effectiveness of the CGCT design strategy that introduces 3D blading to the while ensuring an adequate stable operating range by considering 3D blading in the process of the blade tip during the optimization of the grooves’ axial distribution. CGCT design. 5.1. OptimizationIntegrated Setup 5. CGCT-Blade Optimization The sweep and lean of the blade tip and the axial distribution of the circumferential grooves are Because the CGCT-blade interactions are found to play a significant role in altering the rotor’s optimized together. The total number of design variables is four. The variation ranges of the sweep performance and stability, we undertake an integrated optimization of the CGCT-blade combination. and lean at the blade tip are [−12%, +12%] and [−3.5%, +3.5%], respectively. The ranges of 𝑘 and 𝑠 The goal is to verify the effectiveness of the CGCT design strategy that introduces 3D blading to the controlling the grooves’ location are [−0.8%, +20%] and [1.6%, 12%], respectively. A negative value of blade tip during the optimization of the grooves’ axial distribution. 𝑘 means that the first groove is upstream of the leading edge. The variation range of 𝑠 equals 0.2– 1.5 times the width of the groove. Since the width of each groove is 8% of 𝐶 , , the start position of 5.1. Optimization Setup groove 𝑁 can be expressed by Equation (3): The sweep and lean of the blade tip and the axial distribution of the circumferential grooves are (3) 𝐿 of design = 𝑘 + (8% + 𝑠) ∙ (𝑁 − 1) The variation ranges of the sweep optimized together. The total number variables is four. and lean at the blade tip are [−12%, +12%] and [−3.5%, +3.5%], respectively. The ranges of k and s where 𝑁 ∈ {1,2,3,4,5}. controlling the grooves’ location are [−0.8%, +20%] and [1.6%, 12%], respectively. A negative value of A hybrid aerodynamic optimization algorithm HSADE [36] is used in this paper to search a k means that the first groove is upstream of the leading edge. The variation range of s equals 0.2–1.5 Pareto Front. HSADE combines the differential evolution (DE) algorithm with the radial basis times the width of the groove. Since the width of each groove is 8% of Cax, tip , the start position of function (RBF) response surface. Specifically, DE is a stochastic evolutionary optimization algorithm, groove N canfor be expressed by Equation (3): global searching ability, and its suitability for highwell-known its robustness, its strong dimensional problems [36]. RBF evaluates the similarity between the candidate design variables and Lstart = k + (8% + s)·( N − 1) (3) known samples in the design space to predict their corresponding objective functions by interpolation. The are used to construct the response surface for searching for potential where N ∈ {1, 2, 3, 4,results 5}. optimal individuals. The Pareto Front isalgorithm a group of optimal thatpaper are not dominated by A hybrid aerodynamic optimization HSADE [36]individuals is used in this to search a Pareto any other individuals by the evolution multi-objective optimization. To radial sum up, embedding RBF Front. HSADE combinesobtained the differential (DE) algorithm with the basis function (RBF) response surfaces into the basic DE improves the local searching ability of the basic DE, which helps response surface. Specifically, DE is a stochastic evolutionary optimization algorithm, well-known for to boost the convergence whilesearching obtainingability, an ideal Front. for high-dimensional problems [36]. its robustness, its strong global andPareto its suitability The population size ofbetween each generation is nine. total pressure ratio and the adiabatic RBF evaluates the similarity the candidate designThe variables and known samples in the design efficiency at the design point are set as the two objectives for the optimization. Three operating space to predict their corresponding objective functions by interpolation. The results are used to conditions,the namely, thesurface chokingforpoint, designforpoint, and near-stall point, are computed forFront each construct response searching potential optimal individuals. The Pareto design in the optimization process. The operating conditions, design constraints, and objectives of the CGCT-blade integrated optimization problem are listed in Table 5. Note that the stall mass flow

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is a group of optimal individuals that are not dominated by any other individuals obtained by the multi-objective optimization. To sum up, embedding RBF response surfaces into the basic DE improves the local searching ability of the basic DE, which helps to boost the convergence while obtaining an ideal Pareto Front. The population size of each generation is nine. The total pressure ratio and the adiabatic efficiency at the design point are set as the two objectives for the optimization. Three operating conditions, namely, the choking point, design point, and near-stall point, are computed for each design in the optimization process. The operating conditions, design constraints, and objectives of the CGCT-blade Energies 2018, 11, x FOR PEER REVIEW 17 of 25 integrated optimization problem are listed in Table 5. Note that the stall mass flow of the baseline ND-TAC rotor ND-TAC is 8.67 kg/s, a baseline CGCT can decrease thedecrease stall mass more than of the baseline rotorand is 8.67 kg/s, and a baseline CGCT can theflow stallby mass flow by 0.7 kg/s; near-stall the rotorofwith the CGCT is set at 8.10 Oncekg/s. the more thantherefore, 0.7 kg/s; the therefore, thecondition near-stall of condition the rotor with the CGCT is kg/s. set at 8.10 optimized blade with the CGCT is able to is operate 8.10 kg/s with a total ratio of at least Once the optimized blade with the CGCT able toatoperate at 8.10 kg/s withpressure a total pressure ratio of 1.511, the newly obtained CGCT-blade combination is considered to have a sufficient stall margin. at least 1.511, the newly obtained CGCT-blade combination is considered to have a sufficient stall The purpose of the CGCT-blade optimization is to obtain improvements to Pt0, rat and design margin. The purpose of the CGCT-blade optimization is to obtain improvements to 𝑃η ,at the and 𝜂 at point, as well as a favorable stall margin. the design point, as well as a favorable stall margin. Table 5. Table 5. Definition Definition of of the the CGCT-blade CGCT-blade integrated integrated optimization optimization problem. problem. Working Conditions Constraints Working Conditions Constraints 10.08 kg/s choking point choking point 10.08kg/s kg/s≤≤𝑚 m≤ ≤ 10.34 10.34 kg/s design point = 9.67 kg/s) design point (m (m = 9.67 kg/s) -near-stall point (m = 8.10 kg/s) 1.511 rat≥≥ 1.511 𝑃Pt0, near-stall point (m = 8.10 kg/s) ,

Objectives Objectives - max Pt0,𝑃rat, andand maxmax η max 𝜂 -

-

5.2. 5.2. Results Results and and Analysis Analysis The The optimization optimization ran ran for for 3400 3400 CPU CPU hours hours on on aa cluster cluster with with 300 300 cores cores in in total, total, and and itit converged converged after 32 generations. generations. The Theoptimization optimization histories objectives plotted in Figure 17. after 32 histories of of thethe twotwo objectives are are plotted in Figure 17. The The performance differences between the individuals in the last few generations are small, and performance differences between the individuals in the last few generations are small, and the the objective values concentrated a narrow strip, which indicates optimization objective values are are concentrated on aonnarrow strip, which indicates that that the the optimization has has converged. converged.

(a)

(b)

Figure 17. 17. Convergence Convergence histories histories of of the the two two design design objectives: objectives: (a) (a) the the total total pressure pressure ratio, ratio, and and (b) (b) the the Figure adiabatic efficiency. efficiency. adiabatic

The The values values of of the the two two design design objectives objectives of of the the individuals individuals produced produced in in the the CGCT-blade CGCT-blade optimization are plotted in Figure 18a. Performances of baseline blade 0, swept blade FS, BS, andBS, leaned optimization are plotted in Figure 18a. Performances of baseline blade 0, swept blade FS, and blade and NL, which been discussed in Sectionin4.2, are also in the figure withfigure blue leanedPL blade PL and NL,have which have been discussed Section 4.2,indicated are also indicated in the deltas. Compared with the tested blades, theblades, CGCT-blade integratedintegrated optimization further improves with blue deltas. Compared with the tested the CGCT-blade optimization further the total pressure ratio Pt0, rat and efficiency η. The result verifies effectiveness of improves the total pressure ratiothe 𝑃 adiabatic and the adiabatic efficiency 𝜂 . Thethe result verifies the , effectiveness of the CGCT-blade integrated optimization strategy. The Pareto Front of the optimization is shown in Figure 18b. The difference between these optimal individuals is less than 0.02% for 𝑃 , and no greater than 0.05% for 𝜂.

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the CGCT-blade integrated optimization strategy. The Pareto Front of the optimization is shown in Figure The difference between these optimal individuals is less than 0.02% for Pt0, rat and Energies 18b. 2018, 11, x FOR PEER REVIEW 18 ofno 25 greater than 0.05% for η.

(a)

(b)

Figure18. 18.Design Designobjectives objectives of ofthe theCGCT-blade CGCT-bladeintegrated integratedoptimization optimization results resultsfor for(a) (a)all allqualifying qualifying Figure individuals,and and(b) (b)18 18optimal optimalindividuals individualson onthe thePareto ParetoFront. Front. individuals,

Table Table66shows showsthe theaverage averagedesign designperformance performance of of the the grooved grooved tested tested blades blades and and the the grooved grooved optimized blades on the Pareto Front. The average performance of all of the grooved swept and leaned optimized blades on the Pareto Front. The average performance of all of the grooved swept and blades generatedgenerated for the testfor is slightly than the performance grooved baseline blade leanedmanually blades manually the testlower is slightly lower than the of performance of grooved 0baseline + CT_b.blade Compared with 0 + CT_b,with the CGCT-blade optimization improves Pt0, improves 0 + CT_b. Compared 0 + CT_b, theintegrated CGCT-blade integrated optimization rat and η by 0.53% and 0.29%, respectively. When the optimization results are compared with the solid-casing 𝑃 , and 𝜂 by 0.53% and 0.29%, respectively. When the optimization results are compared with baseline blade, the above two values increase 1.19%increase and 0.90%, respectively. the solid-casing baseline blade, the above twotovalues to 1.19% and 0.90%, respectively. Table Table6.6.The Theaverage averageperformance performanceatatthe thedesign designpoint. point. Type Type

Pt0, 𝑷𝒕𝟎,𝒓𝒂𝒕 rat

η𝜼

SW_0 SW_0 0 +0 CT_b + CT_b tested blades with CT_b tested blades with CT_b CGCT-blade integrated opt.

1.43655 1.43655 1.44596 1.44596 1.44393 1.44393 1.45354

0.83008 0.83008 0.83511 0.83511 0.83437 0.83437 0.83755 0.83755

CGCT-blade integrated opt.

1.45354

For For the the optimal optimal individuals individuals on on the the Pareto Pareto Front, Front, the the average average values values and and variances variances of of the the four four design design variables variables are are shown shown in in Table Table7.7. The Thedifferences differences between between these these optimal optimal individuals individuals are are very very small. small.The The optimal optimalblades bladesare arecharacterized characterizedby byaaforward forwardswept sweptand andpositive positiveleaned leanedtip. tip.The Thedistance distance between two adjacent grooves is approximately 7.7% of C , which is approximately 96% of the between two adjacent grooves is approximately 7.7% of 𝐶ax,tip , , which is approximately 96% of the grooves’ of of k and s into Equation (3), we axial grooves’width. width.By Bysubstituting substitutingthe theaverage averagevalues values 𝑘 and 𝑠 into Equation (3),find we that findthe that the coverage of the CGCT obtained in the optimization is located between 11.7% and 82.5% of C . axial coverage of the CGCT obtained in the optimization is located between 11.7% and ax,tip 82.5% of 𝐶

,

. Table 7. Averages and variances of the values of design variables of the individuals on the Pareto Front.

Table 7. Averages and variances of the values of design variables of the individuals on the Pareto Front. 3D Blading CGCT Distribution Statistics 3D Blading CGCT Distribution k (%Cax,tip ) s (%Cax,tip ) Lean (%(r·`)) Statistics Sweep (%L) average average variance

variance

Sweep (%𝑳) 11.9669 11.9669 0.0419 0.0419

Lean (%(𝒓 ∙ 𝜽)) 2.9675 2.9675 0.3449 0.3449

𝒌 (%𝑪𝒂𝒙,𝒕𝒊𝒑 ) 11.6883 11.6883 0.0028 0.0028

𝒔 (%𝑪𝒂𝒙,𝒕𝒊𝒑 ) 7.7067 7.7067 0.0673 0.0673

The CGCT-blade combination ID 277 marked by the black circle in Figure 18b is taken as the representative of the optimization for further analysis. The values of the sweep, lean, k, and s of ID 277 are 12.0000%, 2.8466%, 11.7790%, and 7.6247%, respectively. For the sake of simplicity, the blade of ID 277 is named “P”, and the CGCT of ID 277 is named “CT_p” in the following discussion.

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The CGCT-blade combination ID 277 marked by the black circle in Figure 18b is taken as the representative of the optimization for further analysis. The values of the sweep, lean, k, and s of ID 277 are 12.0000%, 2.8466%, 11.7790%, and 7.6247%, respectively. For the sake of simplicity, the blade of ID 277 is named “P”, and the CGCT of ID 277 is named “CT_p” in the following discussion. To eliminate the influence of the grid density and numerical errors on the prediction of the performance gains, the fine mesh used to conduct the grid convergence study is generated in the passage of blade P for validation. The design point performance predicted by the medium and fine mesh are compared in Table 8. Although the medium mesh slightly overestimates Pt0, rat and η when compared with the fine mesh, the improvements in Pt0, rat and η predicted by the medium mesh are close to the improvements given by the fine mesh. Therefore, it is proved that the refinement of the grid does not change the trend of the optimization, and the performance improvement predicted by the medium mesh in the optimizer is reliable. The analysis based on results of the medium mesh is performed to further investigate the inner mechanisms. Table 8. Effects of the grid density on the predicted design point performance. Performance and Variations

Medium Mesh (Used in the Optimization)

Fine Mesh (Used for Validation)

Pt0, rat of baseline blade 0

1.43655

1.43449

Pt0, rat of grooved blade 0 + CT_b

1.44596

1.44101

Pt0, rat of optimized grooved blade P + CT_p

1.45361

1.45009

Improvement in Pt0, rat (P+CT_p versus 0)

1.1876%

1.0875%

Improvement in Pt0, rat obtained by optimization (P+CT_p versus 0+CT_b)

0.5291%

0.6301%

η of baseline blade 0

0.83008

0.82853

η of grooved blade 0 + CT_b

0.83511

0.83178

η of optimized grooved blade P +CT_p

0.83753

0.83626

Improvement in η (P+CT_p versus 0)

0.8975%

0.9330%

Improvement in η (P+CT_p versus 0+CT_b)

0.2898%

0.5386%

The speed curves of ID 277 with and without the CGCT are shown in Figure 19. The baseline blade installed with CT_p is also simulated for comparison. As shown by the blue solid squares and the black solid diamonds, the design point performance of the solid casing blade P is almost identical with the performance of the baseline blade, which indicates that the benefits of CGCT-blade integrated optimization are not obtained by improving the solid casing blade’s performance. After installing CGCT, the stall mass flow is reduced below 8 kg/s in both rotors, and the near-stall performance of both are very similar, which shows that the ability of optimized grooves CT_p to extend the stall margin does not vary with the re-shape of the baseline blade. The zoom-in views at the design point show that the performance improvement induced by optimized grooves CT_p in the optimized blade P is larger than the corresponding improvement in the baseline blade. This result demonstrates that the CGCT-blade integrated optimization which alters the 3D shape of the blade and the axial distribution of the CGCT simultaneously, improves the CGCT-blade interactions to pursue an optimal CGCT-blade combination.

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Figure Figure 19. 19. Speed Speed curves curves of of optimal optimal blade blade ID ID 277 277 for for (a) (a) the the total total pressure pressure ratio ratio versus versus the the mass mass flow flow rate; (b) the adiabatic efficiency versus the mass flow rate. rate; (b) the adiabatic efficiency versus the mass flow rate.

The working working capacities capacities and and efficiencies efficiencies of of the baseline baseline blade blade and optimized optimized blade blade with with and The without installing thethe spanwise direction in Figure 20. In 20a, the without installingCT_p CT_pare arecompared comparedalong along spanwise direction in Figure 20.Figure In Figure 20a,total the pressure ratio of optimized blade blade P is reduced only inonly the tip the forward sweep and positive total pressure ratio of optimized P is reduced in region the tip by region by the forward sweep and lean. When installed, the total pressure ratio continues to decrease aboveabove 90% of90% theof span, positive lean.CT_p WhenisCT_p is installed, the total pressure ratio continues to decrease the however, it increases below below 90% of 90% the span. the totalthe pressure ratio improvement induced span, however, it increases of theMoreover, span. Moreover, total pressure ratio improvement by CT_p below 90%below of the span P is obviously the improvement the induced by CT_p 90% in ofoptimized the span blade in optimized bladelarger P is than obviously larger thanin the baseline blade.inFigure 20b shows thatFigure the efficiency improvement induced by CT_p is concentrated in improvement the baseline blade. 20b shows that the efficiency improvement induced by the region above 85% ofinthe Theabove improvement efficiency in optimized blade is also larger CT_p is concentrated thespan. region 85% of of thethe span. The improvement of theP efficiency in than the corresponding improvement the corresponding baseline blade. improvement The comparison thebaseline diffusion factors in optimized blade P is also larger thaninthe in of the blade. The the spanwiseof direction is shown in Figure 20c. In the solid casing is rotor, the forward sweep positive comparison the diffusion factors in the spanwise direction shown in Figure 20c. and In the solid lean inrotor, optimized blade Psweep slightly reduce the lean tip loading; however, when CT_preduce is installed, optimized casing the forward and positive in optimized blade P slightly the tip loading; blade P has a more significant tipoptimized loading reduction than whatsignificant the baseline blade has. Meanwhile, however, when CT_p is installed, blade P has a more tip loading reduction than the enhancement the loading in the non-tip induced by is largerininthe optimized P what the baselineofblade has. Meanwhile, theregion enhancement of CT_p the loading non-tip blade region than that in the baseline blade 0, which indicates that the forward swept and positive leaned blade induced by CT_p is larger in optimized blade P than that in the baseline blade 0, which indicates that tip improves the effectiveness CT_p.blade These plots the demonstrate thatofthe goodThese performance of the forward swept and positiveof leaned tipthree improves effectiveness CT_p. three plots the optimal CGCT-blade the redistributed spanwise loading caused demonstrate that the goodcombination performanceis ofattributable the optimal to CGCT-blade combination is attributable to the by the CGCT-blade interactions in theby tipthe region. Because the optimization not Because change the the redistributed spanwise loading caused CGCT-blade interactions in the tipdoes region. design point does performance of the the spanwise redistribution works to optimization not change the solid designcasing point rotor, performance of the loading solid casing rotor, the spanwise improveredistribution the CGCT-blade interactions, which is helpful to achieve a better overall performance of the loading works to improve the CGCT-blade interactions, which is helpful to achieve a CGCT-blade better overallcombination. performance of the CGCT-blade combination. Figure 20a 0+ and P+ is Figure 20a shows shows that that the the difference differenceininthe thetotal totalpressure pressureratio ratiobetween between 0 +CT_p CT_p and P CT_p + CT_p remarkable at 85% of the span; thus, the static pressure (P ) distributions at the corresponding position is remarkable at 85% of the span; thus, the static pressure (𝑃 ) distributions at the corresponding s are plotted Figure in 21 Figure for comparison. The forward sweep and positive leanpositive in the tip of blade push position areinplotted 21 for comparison. The forward sweep and lean in thePtip of the shockwave leading to a leading larger aft-loading in chordwise direction. After installing blade P push thedownstream, shockwave downstream, to a larger aft-loading in chordwise direction. After CT_p, Ps on the pressure is increased. Compared with CT_p in the baseline theblade, zoom-in installing CT_p, 𝑃 on theside pressure side is increased. Compared with CT_p in the blade, baseline the view in this shows CT_p inCT_p the optimized blade isblade moreiseffective at increasing Ps on the zoom-in viewfigure in this figurethat shows that in the optimized more effective at increasing 𝑃 pressure side between 50% and50% 95%and of the chord length. thatNote Ps in that this section of the optimized on the pressure side between 95% of the chordNote length. 𝑃 in this section of the blade with blade a solidwith casing equals Ps ofequals the baseline blade with ablade solid with casing, which means the increase optimized a solid casing 𝑃 of the baseline a solid casing, which means of Pincrease is further improved after the CGCT-blade the ofby 𝑃 CGCT induced by CGCT is further improved after the optimization. CGCT-blade optimization. s induced

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(a) (a)

(b) (b)

(c) (c) Figure 20. Spanwise distribution of (a) the total pressure ratio, (b) the adiabatic efficiency, and (c) the Figure 20. 20. Spanwise Spanwise distribution and (c) (c) the the Figure distribution of of (a) (a) the the total total pressure pressure ratio, ratio, (b) (b) the the adiabatic adiabatic efficiency, efficiency, and diffusion factor at the design point for optimal blade P and baseline blade 0 with and without CT_p. diffusion factor factor at at the the design design point point for for optimal optimal blade blade PP and and baseline baseline blade blade 00 with with and and without without CT_p. CT_p. diffusion

Figure 21. Static pressure atat85% Figure pressuredistribution distribution 85%ofof ofthe thespan spaninin inbaseline baselineblade blade000and andoptimized optimizedblade blade“P” “P” Figure 21. 21. Static Static pressure distribution at 85% the span baseline blade and optimized blade “P” with and without CT_p. with and without CT_p. with and without CT_p.

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At 93% 93% of of the the span, span, flow flow structures structures of CT_p and and P P ++ CT_p CT_p are are shown shown in in Figure Figure 22. 22. The The main main At of 00 + + CT_p differences between the two flow fields are illustrated in this figure. The detached bow shock in the differences between the two flow fields are illustrated in this figure. The detached bow shock in the flow passage passage of of blade blade PP that that has has aa forward forward swept swept and and positive positiveleaned leanedtip tipisiscloser closerto tothe thedownstream. downstream. flow The area of the shock-induced separation on the suction side (SS) of blade P is also smaller. Moreover, The area of the shock-induced separation on the suction side (SS) of blade P is also smaller. Moreover, because CGCT in blade P is more effective at relieving the blockage than CGCT in the baseline blade because CGCT in blade P is more effective at relieving the blockage than CGCT in the baseline blade is, the the flow flow tube tube sandwiched sandwiched between between PS PS and and the the blockage blockage expands, expands, which which decelerates decelerates the the near-wall near-wall is, flow inside the tube and increases the static pressure in the aft-part of the blade tip (see Figure 21). flow inside the tube and increases the static pressure in the aft-part of the blade tip (see Figure 21). To sum up, when when the the airfoil airfoil is is more more aft-loaded aft-loaded in in chordwise chordwise direction direction due due to to the the forward forward sweep sweep and and To sum up, positive lean, the effectiveness of CGCT to enhance the working capacity of the aft-part of the blade positive lean, the effectiveness of CGCT to enhance the working capacity of the aft-part of the blade is improved. improved. is

(a)

(b)

Figure Figure 22. 22. Comparison Comparison of of flow flow structures structures between between the the baseline baseline blade blade with with CT_p CT_p and and the the optimized optimized blade blade with with CT_p CT_p at at the the design design point: point: (a) (a) baseline baseline blade blade with with optimized optimized CGCT: CGCT: 00 ++ CT_p; CT_p; (b) (b) optimized optimized CGCT-blade combination: P + CT_p. CGCT-blade combination: P + CT_p.

6. Conclusions Conclusions 6. This paper paper studies studies multiple multiple circumferential circumferential casing casing grooves’ grooves’ axial axial distribution distribution design design and and This optimization. The conventional CGCT design with a given blade is developed into a CGCT-blade optimization. The conventional CGCT design with a given blade is developed into a CGCT-blade integrated optimization. optimization. InInthe the process process of of the the optimization, optimization, the the performance performance of of CGCT-blade CGCT-blade integrated combinations isisevaluated evaluatedatatthe thechoking chokingpoint, point, design point near-stall point, which fulfills combinations design point andand near-stall point, which fulfills the the constraints while improving the objectives. The work can be summarized as follows: constraints while improving the objectives. The work can be summarized as follows: (1) A CGCT-blade combination with good design point performance does not necessarily results (1) Anear-stall CGCT-blade combination design point does not necessarily in good performance. Forwith the good ND-TAC rotor, theperformance most effective CGCT that leadsresults to the in good near-stall performance. For the ND-TAC rotor, the most effective CGCT that leads to and the greatest working capacity improvement in the blade at the design point is located between 40% greatest capacity improvement in the blade at the designresults point is between and 100% of working the tip chord length; however, this CGCT configuration inlocated the smallest stall40% margin 100% of the tip chord length; however, this CGCT configuration results in the smallest stall margin enhancement, because the grooves near the trailing edge are ineffective at suppressing the growth of enhancement, grooves near the trailing edgethat are ineffective at suppressing the growththe of the blockage inbecause the tip the region. Therefore, it is suggested the CGCT design should coordinate the blockage in the tip region. it is suggested that the CGCT design should coordinate the effects of CGCT under both theTherefore, design and near-stall conditions. effects of CGCT under both the design and near-stall conditions. (2) The sweep and lean introduced to the tip alter the CGCT-blade interactions, leading to the (2) The sweep and The leanworking introduced to the tipblade alter at thedifferent CGCT-blade interactions, leading to the performance variations. ability of the span, as well as the fluid exchange performance variations. workingare ability of the blade at different as well astothe fluida between the passage and The the grooves, affected by the 3D blading, thus,span, it is promising obtain exchange between the passage and the grooves, are affected by the 3D blading, thus, it is promising better CGCT-blade combination if the sweep and lean are introduced in the process of the CGCT design. to obtain a better CGCT-blade combination if the sweep and lean introduced in thethe process ofand the (3) The CGCT-blade integrated optimization improves theare coupling between CGCT CGCT design. the blade tip, which strengthens the ability of the CGCT in improving the rotor’s design point (3) The CGCT-blade integrated optimization improves the coupling between the CGCT and the blade tip, which strengthens the ability of the CGCT in improving the rotor’s design point performance. Compared with the baseline blade and all of the tested re-stacked blades with CGCTs,

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performance. Compared with the baseline blade and all of the tested re-stacked blades with CGCTs, the CGCT-blade integrated optimization achieves the highest total pressure ratio and adiabatic efficiency while maintaining a sufficient stall margin. In the optimal CGCT-blade combination, a forward swept and positive leaned tip are generated, together with a CGCT located between 11.7% and 82.5% of Cax,tip . In the spanwise direction, the blade loading is reduced in the tip region. In the chordwise direction, the detached bow shock is pushed downstream and the blade is more aft-loaded. These variations in loading are good for the CGCT to reduce the blockage and enhance the working capacity of the non-tip span and aft-part of the blade. The newly developed CGCT-blade integration optimization method is helpful for advancing the understanding of CGCT-blade interactions. The combination of the CGCT and the 3D blading technique breaks the boundary between the CGCT design and the blade design. However, this design strategy is only verified in the ND-TAC rotor with the assistance of CFD. Efforts are needed to validate the effectiveness of this method in other types of rotors by both CFD calculations and experiments. In future work, the CGCT-blade integrated design and optimization method will be refined by introducing more design variables to increase the design freedom of both the blade and CGCT. Author Contributions: Formal Analysis, Software, Writing–Review & Editing, W.S.; Methodology, Data Curation, Project Administration, Y.Z.; Conceptualization, Resources, Supervision, H.C. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 11872230). Acknowledgments: The authors would like to thank the Notre Dame Turbomachinery Laboratory for providing the geometry and the experimental data of the ND-TAC. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature Nc Utip Nb Rcasing Rhub τ m Ms Pt0, rat Tt0, rat η Marel Pt0, rel Df Wr Cax Ps PS SS

Rotational speed (rpm) Blade tip velocity (m/s) Blade number Radius of casing (mm) Radius of hub (mm) Tip gap size (mm) Mass flow (kg/s) Stall mass flow (kg/s) Total pressure ratio Total temperature ratio Adiabatic efficiency Relative Mach number Non-dimensional relative total pressure Diffusion factor Relative radial velocity (m/s) Axial chord length (mm) Static pressure (Pa) Pressure side Suction side

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