1 Introduction

Airflow through tip clearance region in an axial compressor is a primary element affecting on the stall and surge phenomena, and thus is a key concern in researches to improve the compressor performance. Many experimental and numerical researches that based on controlling for the airflow in the tip leakage region have been performed. The bleeding airflow is one of the approaches to improve the compressor performances. Andrew [1] determined the changes in performance of F404-GE-400 afterburning turbofan engines with the application of flow extraction in the F/A-18 High Alpha Research Vehicle at the NASA Dryden Flight Research Facility. The experimental result indicated that the amount of thrust generated was linearly reduced with a rise in bleeding flow at all power settings. Hathaway [2] introduced a numerical passive control method to deal with the detrimental endwall blockage via bleeding flow in a low-speed axial compressor rotor at 8750 rpm. The maximum enhancement in stable range extension was 55% when the bleeding position was at 70% of rotor chord length with a bleeding flow rate of 3.5% of choking mass flow rate; however, the issue in this research was that the results from numerical simulation and experiment differed considerably from each other for the case of smooth casing. Wellborn and Michael [3] examined the influences of axisymmetric bleed slot position on compressor’s blades shroud surface of a single-stage subsonic axial compressor. The results with 5% bleed rate illustrated an improving adiabatic efficiency by 0.25% on rotor shroud surface and 0.4% on stator shroud surface.

Ress et al. [4] investigated a bleeding system located on casing region downstream of rotor in a multistage axial compressor. Numerical simulation indicated that the efficiency and stall margin were increased 1% and 5%, respectively, with a bleed flow rate of 3%. Gomes et al. [5] experimented the two and four bleeding channels located downstream stator vanes from 8 to 12% of core airflow. The measurement using Doppler-global-velocimetry (DGV) and steady pneumatic measurement showed that bleed slot lip design provided greater discharge coefficient than that of flush design. A research from Leishman and Cumpsty [6] investigated a bleed shape geometry with endwall ramps integrated into the blade passage in the same stator configuration to compare the experimental and numerical results for ramp and holes type of bleeding design. It was showed that the static pressure coefficient obtained from real test and simulation was matched for ramp type, but the experimental result of the coefficient was superior to numerical one in the case of holes type. Steady simulation and experiment carried out by Ponick et al. [7] showed the reduction losses and increment in pressure ratio of a low-speed compressor using bleed air ejection near stator shroud suction surface. Peltier et al. [8] presented an airflow bleeding system located downstream stator casing wall with varied bleeding rates ranging from 4 to 12% of primary inlet mass flow rate in an axial compressor, SGT5-4000F Siemens. The experiment data showed that the static pressure ratio rose from 1.029 to 1.136 with an increase of bleeding ratio from 4 to 10%, whereas the total pressure loss was increased from 0.1125 to 0.1537. Zhao et al. [9] and Cui et al. [10] presented the effect of air ejection close to stator corner from 50 to 100% of design rotation speed in a single-stage transonic axial compressor, NASA Stage 35. The results showed that the highest increase at 100% rotational speed was 0.79% at 4% and 4.5% of bleeding rate. At 80% rotational speed, the maximum efficiency was also improved and reached highest increase of 0.85% at the 4% bleeding rate, while the stall margin extended by 39.8% relating to the smooth casing. Grimsha et al. [11] provided the influences of nonuniform air bleeding on the stall margin of a low-speed axial compressor. The results showed that the total pressure increased at near-stall condition and stall margin rise slightly with an increment of bleed rate from 2.1 to 6.2% of primary core mass flow as compared to without bleed rate. Dinh et al. [12] presented the numerical study of the effects of a circumferential bleeding channel’s position as well as bleed mass flow rate on aerodynamic performance of a single-stage transonic axial compressor, NASA Stage 37. Compared to the smooth casing, the maximum of stall margin using bleeding method increased the stall margin by a maximum increment of approximately 1.31%.

Lerche et al. [13] suggested one-way FSI method to predict stresses on the blades in a single-stage centrifugal compressor, good agreement between numerical and experimental results from a real test was obtained. Wang et al. [14] introduced the one-way FSI analysis for a 2-MW offshore wind turbine blades to determine crucial parameter for structural analysis, including the torque value, stress and strain, for the full machine, and these values were compared with other researcher results, such as Chen et al. [15]. Song et al. [16] compared the total pressure ratio, efficiency and stress between optimum and initial design of a transonic single-stage axial compressor, NASA Rotor 37, using one-way FSI. The results showed that the adiabatic efficiency increases by 0.3% with a constant of total pressure ratio and the maximum stress decreases to 0.9 MPa. However, the author did not mention the influence of the rotation speed of rotor on the stress calculation. Kang and Kim [17] presented an optimum impeller design of a centrifugal compressor using FSI analysis and response surface method for optimization method. The results showed that the efficiency increase by 1% and the stress decreases by 10% with a constant of pressure ratio for optimum design as compared to initial design.

In the present work, a circumferential rotor-bleeding ejector, situated from rotor leading and trailing edge on rotor shroud surface, in combination with a low bleed mass flow rate (smaller than 1% of the main mass flow at choking condition of the smooth casing), was examined to determine its effects in the aerodynamic and structural performances of a transonic single-stage axial compressor. To prove the performance of a single-stage transonic axial compressor using bleeding airflow, the aerodynamic performance was evaluated by solving three-dimensional (3-D) Reynolds-averaged Navier–Stokes (RANS) equations. And then, the structural performance of rotor blades is determined using ANSYS-CFX pressure load on pressure and suction sides of rotor and stator blades. The aerodynamic and structural performances of a single-stage transonic axial compressor were also investigated for various geometric parameters of circumferential rotor-bleeding ejector (bleeding and ejection angles, bleeding depth at ejection surface, bleeding thickness of mechanical support, bleeding position, bleeding radii curvature, and bleeding width) and the ejection mass flow rate. The aerodynamic and structural performances results (total pressure ratio, adiabatic efficiency, stall margin, stable range extension, Von-Mises stress, and total deformation on rotor and stator blades) of this parametric study were compared to the results of the smooth casing without bleeding air.

2 Numerical Analysis

2.1 Compressor Model

The transonic axial compressor used in this investigation is a single-stage axial compressor, NASA stage 37, where the rotor blades’ type is 36 NASA Rotor 37 blades rotating at a speed of 17,185.7 rpm for 100% design speed and the stator blades type are 46 NASA Stator 37 stator blades. The specification design of NASA stage 37 was described by Reid and Moore [18], as shown in Table 1 and detailed by Dinh et al. [19, 20]. Figure 1a shows the 3D view of circumferential rotor-bleeding ejector in a single-stage transonic axial compressor. The axial compressor geometry and parameters’ geometric definition of the circumferential rotor-bleeding ejector are shown in Fig. 1b, where the circumferential rotor-bleeding ejector is circumferential and placed on the rotor shroud surface in the middle of leading and trailing edges. The bleed angle (α) indicated the angle between bleeding airflow and rotor shroud surface along the flow direction. The ejection angle (β) illustrated the angle between ejection airflow and rotor shroud surface along the opposite direction of airflow at ejection position. The depth of the ejection port (D) was determined, which was non-dimensionalized by rotor tip clearance (CR); while the bleed port is designed also by the thickness (H) of mechanical support, this value is compared with rotor tip clearance (τ). The circumferential rotor-bleeding position (L) indicated the circumferential rotor-bleeding position that compared to the rotor tip leading edge and the bleeding radii curvature (R) indicated the curvature value of circumferential rotor-bleeding casing connected from bleeding to ejection. The widths of circumferential rotor-bleeding port (W) measured with reference to rotor tip chord length (CR) were also tested. The ejection mass flow rate (\( \dot{m}_{\text{B}} \)) was also non-dimensionalized by airflow at compressor inlet of the smooth casing (\( \dot{m}_{\text{SC}} \)) and selected as an operating parameter.

Table 1 NASA stage 37 design specifications [18]
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Fig. 1
figure 1

NASA stage 37 geometry and computational domains with rotor-bleeding airflow: a 3D view, b meridional plane view

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2.2 Performance Parameters

The compressor aerodynamic performance parameters are the total pressure ratio (PR), adiabatic efficiency (η), stall margin (SM), and stable range extension (SRE), which are described by Dinh et al. [19, 20], as shown in the following:

$$ {\text{PR}} = \frac{{P_{\text{t,out}} }}{{P_{\text{t,in}} }}, $$
(1)
$$ \eta = \frac{{\left( {\frac{{P_{\text{t,out}} }}{{P_{\text{t,in}} }}} \right)^{{\frac{\gamma - 1}{\gamma }}} - 1}}{{\left( {\frac{{T_{\text{t,out}} }}{{T_{\text{t,in}} }}} \right) - 1}}, $$
(2)
$$ {\text{SM}} = \left( {\frac{{\dot{m}_{\text{peak}} }}{{\dot{m}_{\text{stall}} }} \times \frac{{PR_{stall} }}{{{\text{PR}}_{\text{peak}} }} - 1} \right) \times 100\% , $$
(3)
$$ {\text{SRE}} = \left( {\frac{{(\dot{m}_{\text{max} } - \dot{m}_{\text{stall}} )_{\text{bleeding}} - (\dot{m}_{\text{max} } - \dot{m}_{\text{stall}} )_{\text{smooth}} }}{{(\dot{m}_{\text{max} } - \dot{m}_{\text{stall}} )_{\text{smooth}} }}} \right) \times 100\% , $$
(4)

where \( \dot{m}_{\text{peak}} \), \( \dot{m}_{\text{stall}} \), and \( \dot{m}_{\text{max} } \) are the mass flow rates at peak efficiency, near-stall, and choke conditions, respectively. \( {\text{PR}}_{\text{peak}} \) and \( {\text{PR}}_{\text{stall}} \) are the total pressure ratio at peak efficiency and near-stall conditions, respectively. γ, \( P_{\text{t}} \), and \( T_{\text{t}} \) indicate the specific heat ratio, total pressure, and total temperature, respectively. For the smooth casing without bleeding airflow, the stable range extension value was begin at zero. The structure performance parameters of the axial compressor are considered as Von-Mises stress and total deformation of NASA stage 37 blades (rotor and stator blades).

2.3 Numerical Analysis

In this work, three-dimensional RANS equations using the k-ε turbulence model with the scalable wall function were solved numerically for the aerodynamic analysis. The FSI analysis is used to determine the Von-Mises stress near rotor hub surface and total deformation on rotor tip leading edge. The computational domain is shown in Fig. 1b, where one NASA Rotor 37 and one NASA Stator 37 with a rotor-bleeding ejector located on the rotor shroud surface between leading and trailing edge. The mesh type using in this research is hexahedral elements, where the O-type grid, H/J/C/L-type grids were used near the rotor and stator blades and in the other regions of the rotor and stator blocks, respectively, and the H-type grid was applied for the bleeding ejector, as shown in Fig. 2a. The compressor blades structure mesh was using H-type grid, as shown in Fig. 2b, where the mesh on the compressor blades of structure and flow parts was coincided. The flow analysis was used the commercial ANSYS-CFX 15.0 [21], where the rotor and stator blades shape designs and generate the rotor and stator computational grid which were used in Blade-Gen and Turbo-Grid, respectively. The rotor-bleeding ejector and grid were utilized Design-Modeler and ICEM-CFD, respectively. ANSYS CFX-Pre, CFX-Solver, and CFX-Post were utilized to define boundary conditions, solve governing equations, and to post-process the results, respectively. For the structure analysis, the compressor blades (rotor and stator blades) were exported from Design-Modeler into ANSYS Multiphysics, which were used also to mesh, define boundary conditions, solve governing equations, and to post-process the results, respectively.

Fig. 2
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Fluid and structure of the grid system: a flow mesh, b structure mesh

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The boundary conditions of this research are used as the same in Dinh et al. [19, 20] and the connection interfaces between the stator and rotor domains and between rotor shroud surface and rotor-bleeding ejector domain were selected the general grid interface (GGI) method. The k-ε turbulence model with y + values in the range of 20–100 at the first nodes near the wall and scalable wall function were used to solve the working fluid equations, where Dinh et al. [19, 20] showed the good agreement between numerical and experimental results for a single-stage transonic axial compressor, NASA stage 37. For structure analysis of compressor blades [22, 23], the titanium alloy with the density of 4620 kg/m3, tensile and compressive yield strength of 930 MPa, and tensile ultimate strength of 1070 MPa were used for rotor blades, whereas the structural steel [24, 25] with the density of 7850 kg/m3, tensile and compressive yield strength of 250 MPa and tensile ultimate strength of 460 MPa were used for stator blades, which are implanted in the ANSYS Multiphysics. The rotor hub surface was rotational velocity at rotational speed, and the stator shroud surfaces were fixed, as shown in Fig. 2b.

The last converged point, where the total pressure ratio reaches the maximal value, was determined as the near-stall condition using the convergence criteria by Chen et al. [26]. The performance curves were constructed by increasing the average static pressure at the outlet surface from the choking condition (0 Pa) to the last stable convergence point, as presented in Dinh et al. [19, 20]. The PC with an Intel® Core I7-4930 K CPU@3.40 GHz 32.0 GB was used to perform the computation with seven cores for each calculation [27]. The computational time for one case was about 6 h with the optimum grid for booth fluid and structure simulations, and it was total 160 h in computational time for one design parameter.

3 Results and Discussion

3.1 Grid-Dependency Test and Validation

The experimental data of a single-stage transonic axial compressor, NASA stage 37, reported by Reid and Moore [18], were used to validate the numerical simulation. A grid dependency and validation tests were performed for the single-stage transonic axial compressor, as shown in Fig. 3, where Fig. 3a shows the total pressure ratio and adiabatic efficiency curves of four grid system structures. Figure 3b presents the performance curves of total pressure ratio and adiabatic efficiency obtained from numerical simulation of the single-stage transonic axial compressor for the case of smooth casing without rotor-bleeding airflow is compared to the experimental data reported by Reid and Moore [18]. The results show that the numerical simulation was able to closely reproduce the data from their experiment. The predicted peak adiabatic efficiency is 83.85%, which is only slightly lower than the measurement, at 84.00%. In addition, the predicted total pressure ratio at peak efficiency condition is 2.0045, which is very close to the experimental result of 2.000. In simulation, the compressor reached near-stall condition at 93.85% of the main choking mass flow rate, which is very close to the measured result, 93.65%. The predicted stall margin and stall margin obtained from real test are also very similar, 9.95% compared to 10.00%.

Fig. 3
figure 3

Grid dependency tests and validation of smooth casing NASA stage 37: a total pressure ratio and adiabatic efficiency curves, b validation between numerical and experimental results

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3.2 Reference Rotor-Bleeding Airflow

To examine the effects of a circumferential rotor-bleeding ejector combined with airflow ejection on the performance of a single-stage transonic axial compressor, NASA stage 37 were investigated using 3-D Reynolds-averaged Navier–Stokes (RANS) equations. Table 2 describes the reference design of the rotor-bleeding ejector combined with airflow ejection, where the bleeding angle (α) and the ejection angle (β) of the bleeding ejector are 45° and 0°, respectively, and all others parameter is non-dimensionalized.

Table 2 Reference design of rotor-bleeding airflow
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A range of node number varied from 10,000 to 55,000 was evaluated for the grid of the reference rotor-bleeding airflow. From the results shown in Fig. 4a, the optimum number of nodes for the reference rotor-bleeding airflow of 36,800 was chosen for all further calculations. The results in Fig. 4b present a large extension of the stable range extension as the rotor-bleeding airflow delayed the near-stall condition from 0.9385 (smooth casing) to 0.9250 (reference rotor-bleeding airflow design) with a small rise in adiabatic efficiency (83.94% compared to 83.85% for the smooth casing). At peak adiabatic efficiency condition, the reference design provided a superior total pressure ratio to that of the smooth casing (2.0148 compared to 2.0045). The reference rotor-bleeding airflow design extend the stall margin to 11.26%, which is 13.17% higher than that of the smooth casing (9.95%), and the stable range extension increases significantly from 0.00% for the smooth casing to 21.92% for the reference design.

Fig. 4
figure 4

NASA stage 37 performance curves with rotor-bleeding airflow: a rotor-bleeding ejector grid dependency tests at peak condition, b total pressure ratio and adiabatic efficiency curves

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At peak adiabatic efficiency condition, Fig. 5 illustrates the Mach number contours on 98% span surface of NASA stage 37. The low-speech zones, corresponded to Mach number of 0.4 existing near rotor pressure leading edge area and between stator pressure and suction surfaces (Fig. 5a), were largely reduced with the presence of circumferential rotor-bleeding airflow, as shown in Fig. 5b. This reduction of low-speech zones contributes to the improvement of the adiabatic efficiency with a circumferential rotor-bleeding airflow as compared to the smooth casing. Figure 6 shows the equivalent Von-Mises stress at peak efficiency condition for NASA stage 37, where the maximal equivalent Von-Mises stress using rotor-bleeding airflow on rotor and stator blades is slightly higher than of the smooth casing, 399.9 MPa, 84.103 MPa and 399.63 MPa, 83.55 MPa, respectively. Whereas Fig. 7 shows the total deformation at peak efficiency condition for NASA stage 37, where the maximal total deformation using rotor-bleeding airflow on rotor and stator blades is smaller than of the smooth casing, 1.2998 mm, 0.10466 mm, and 1.3051 mm, 0.10225 mm, respectively (the rotor tip clearance is 0.4 mm).

Fig. 5
figure 5

Relative Mach number contours on 98% span surface of NASA stage 37 at peak adiabatic efficiency condition using rotor-bleeding airflow: a smooth casing, b reference rotor-bleeding airflow

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Fig. 6
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Equivalent Von-Mises stress at peak efficiency condition for NASA stage 37 using rotor-bleeding airflow: a smooth casing, b reference rotor-bleeding airflow

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Fig. 7
figure 7

Total deformation at peak efficiency condition for NASA stage 37 using rotor-bleeding airflow: a smooth casing, b reference rotor-bleeding airflow

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As illustrated in Fig. 8, the rotor suction length of reattachment lines was shorten from 85% span for the smooth casing to 72% span for the reference rotor-bleeding airflow, while the separation lines were slightly bent downward when a circumferential rotor airflow bleeding was presented. A combination of these reduction of rotor tip leakage vortex, separation, and reattachment lines provides the reasons for the enhancement of stall margin and stable range extension of a single-stage transonic axial compressor when a circumferential rotor-bleeding airflow is utilized.

Fig. 8
figure 8

Streamlines on NASA stage 37 blades at near-stall condition using rotor-bleeding airflow: a smooth casing, b reference rotor-bleeding airflow

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3.3 Parametric Study

The aerodynamic and structural performances of a single-stage transonic axial compressor were investigated for a wide range of geometric parameters of rotor-bleeding airflow ejector (bleeding and ejection angles, bleeding height at ejection surface, bleeding thickness, bleeding position, bleeding radii curvature, and bleeding width at bleeding surface on rotor shroud surface) and the ejection mass flow rate, as shown in Table 3. The parametric study results focusing on these eight parameters of the circumferential rotor-bleeding airflow (Table 3) on aerodynamic and structural performances of NASA stage 37 are illustrated in Tables 4, 5, 6, 7, 8, 9, 10, and 11

Table 3 Parametric study of parameter ranges for rotor-bleeding airflow
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Table 4 Effect of bleeding angle (α°) on aerodynamic and structural performances for NASA stage 37
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Table 5 Effect of bleeding angle (β°) on aerodynamic and structural performances for NASA stage 37
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Table 6 Effect of ejection depth (D) on aerodynamic and structural performances for NASA stage 37
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Table 7 Effect of bleeding thickness (H) on aerodynamic and structural performances for NASA stage 37
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Table 8 Effect of bleeding position (L) on aerodynamic and structural performances for NASA stage 37
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Table 9 Effect of bleeding radii curvature (R) on aerodynamic and structural performances for NASA stage 37
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Table 10 Effect of bleeding width (W) on aerodynamic and structural performances for NASA stage 37
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Table 11 Effect of ejection mass flow rate (α°) on aerodynamic and structural performances for NASA stage 37
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In the parametric study, the values of the parameters except that being tested were fixed as the reference values. Table 4 shows the effects of bleeding angle, which provides a higher the total pressure ratio and adiabatic efficiency compared to those of the smooth casing when the bleeding angle increases with the exception of 30°. The stall margin and stable range extension are also improved when bleeding angle is increased from 30 to 75°, and the highest values reach 11.74 and 32.21%, respectively, at bleeding angle of 60°. The Von-Mises stress is almost constant for all case, the maximal rise in Von-Mises stress is only 0.11% at bleeding angle of 60°, whereas the maximal reduction in total deformation at rotor tip blades is 1.66% at bleeding angle of 30° in comparison with the smooth casing. As shown in Table 5, the total pressure ratio and adiabatic efficiency are greatly enhanced from 2.0148 to 2.0195 and 83.94 to 84.11%, respectively, as the ejection angle increase from 0° to 180°. Meanwhile, with the same change in ejection angle, the stall margin and stable range extension experience a fall from 11.26 to 9.80% and from 21.92 to 3.55%, respectively. The Von-Mises stress is almost constant for all case (the maximal rise in Von-Mises stress is only 0.11%, at ejection angle of 180° as compared to the smooth casing), whereas the maximal total deformation at rotor tip blades is reduced by 0.41%, at ejection angle of 0°.

The total pressure ratio, efficiency, and Von-Mises stress are least sensitive to the ejection depth, as shown in Table 6, where the maximal difference values of total pressure ratio and adiabatic efficiency are 0.58, 0.1 and 0.08%, respectively, as compared to those of the smooth casing. The most influence of ejection depth is in stall margin, stable range extension, and total deformation, and the maximal difference values compared to those of the smooth casing are 1.52, 22.43, and 0.76%, respectively. Table 7 indicates that the bleeding thickness has a slight influence on the total pressure ratio and adiabatic efficiency, and the maximal increase values are only 0.65 and 0.09%, respectively, as compared to the smooth casing. The stall margin and stable range extension are highly raised 1.78 and 29.01% at 100% of rotor blade tip clearance, respectively, as those of the smooth casing, and they are decreased with an increase in bleeding thickness. The maximal value in Von-Mises stress is almost constant for all variation of bleeding thickness; however, the total deformation at rotor tip blades is decreased to 0.49 at 500% of rotor blade tip clearance as compared to the smooth casing.

The bleeding position plays a major role in all performance, as shown in Table 8, where each maximum performance value is not the same; 2.0148 at 40% of rotor blade tip chord length for total pressure ratio, 84.08% at 60% for adiabatic efficiency, 12.18% at 50% for stall margin, and 21.92% at 40% for stable range extension, respectively. The maximal value in Von-Mises stress is 399.91 MPa (only 0.07%) at 60% of rotor blade tip chord length, while the maximal total deformation at rotor tip blades is reduced by 1.62% at 50% of rotor blade tip chord length as compared to the smooth casing. In Table 9, it is shown that the bleeding radii curvature of circumferential rotor-bleeding ejector has slight effects on compressor’s performance, as the total pressure ratio and adiabatic efficiency are slightly enhanced with bleeding radii curvature varied from 2.5 to 10% of rotor blade tip chord length, and achieve the maximum values of 2.0163 and 83.97, respectively, at 2.5%. The stall margin and stable range extension are largely increased from 10.00 to 11.18% and 3.94 to 19.14%, respectively, as the bleeding radii curvature augments from 2.5 to 10%, and attain the highest values of 11.26 and 21.92% at bleeding radii curvature of 5.0%. The maximal value in Von-Mises stress is 399.96 MPa (only 0.08%) at 2.5% of rotor blade tip chord length, whereas the maximal reduction in total deformation at rotor tip blades is 0.45% at 10% of rotor blade tip chord length as compared to the smooth casing.

The bleeding width is one of the most crucial factors to the compressor’s aerodynamic performances and deformation, except on Von-Mises stress, as shown in Table 10. The total pressure ratio and adiabatic efficiency reach the maximum values of 2.0182 and 84.12%, respectively, at 3% of rotor blade tip chord length, and the highest stable range extension of 22.52% is reached at 6%, whereas the stall margin is 12.07% at 7% of rotor blade tip chord length. The maximal value in Von-Mises stress is 400.02 MPa (only 0.1%) at 3% of rotor blade tip chord length, whereas the maximal reduction in total deformation at rotor tip blades is 1.71% at 7% of rotor blade tip chord length as compared to the smooth casing. As shown in Table 11, the total pressure ratio, adiabatic efficiency, equivalent Von-Mises stress, and total deformation are gradually augmented from 2.0055 to 2.0247, 83.75 to 84.06%, 399.58 MPa to 400.25 MPa, and 1.2942 mm to 1.3078 mm, respectively, when the ejection mass flow rate varied from 0.3 to 0.7% of smooth casing. The stall margin and stable range extension are increased when the ejection mass flow rate rises from 0.3 to 0.5% and decreases when the ejection mass flow rate continually increases, the maximal values are 11.26 and 21.92%, respectively, at 0.5% ejection mass flow rate.

The results in Tables 4, 5, 6, 7, 8, 9, 10, and 11 show that the maximal equivalent Von-Mises stress on rotor and stator blades are 400.25 MPa and 85.06 MPa for the maximum total pressure ratio (\( \frac{{\dot{m}_{\text{B}} }}{{\dot{m}_{\text{SC}} }} = 0.7\% \)), which correspond to 0.16 and 1.81% increasing as compared to the smooth casing results (399.63 MPa and 83.55 MPa). The minimum total deformation on rotor and stator tip blades are 1.284 mm and 0.10105 mm for the maximum stall margin (\( \frac{L}{{C_{\text{R}} }} = 50\% \)), which reduce by 1.62 and 1.17%, respectively, as compared to the corresponding of the smooth casing results (1.3051 mm and 0.10225 mm).

In Fig. 9, the contours of relative Mach number at 98% span surface at peak efficiency condition of maximum adiabatic efficiency (\( \frac{W}{{C_{\text{R}} }} = 3\% \)) and maximum total pressure ratio (\( \frac{{\dot{m}_{\text{B}} }}{{\dot{m}_{\text{SC}} }} = 0.7\% \)) are compared to those of the reference design and smooth casing. The results show that the size of low-speed zones in rotor and stator domains existed in the smooth casing (Fig. 5a) has been reduced with the application of circumferential rotor-bleeding ejector (Fig. 9a, b). However, the reference design (Fig. 5b), maximum adiabatic efficiency (\( \frac{W}{{C_{\text{R}} }} = 3\% \)), and maximum total pressure ratio (\( \frac{{\dot{m}_{\text{B}} }}{{\dot{m}_{\text{SC}} }} = 0.7\% \)) are almost the same. The streamlines on NASA stage 37 blades of maximum stall margin (\( \frac{L}{{C_{\text{R}} }} = 50\% \)) and maximum stable range extension (\( \alpha = 60^\circ \)) at near-stall condition are compared to those of the reference rotor-bleeding airflow and smooth casing, as shown in Fig. 10. The results indicate that the reattachment line on the rotor blade suction surface of maximum stable range extension (\( \alpha = 60^\circ \)) is shortest as compared to those of the maximum stall margin (\( \frac{L}{{C_{\text{R}} }} = 50\% \)), reference rotor-bleeding airflow, and smooth casing, 68% span, 85% span, 72% span, and 85% span, respectively, which justifies the maximum stable range extension at \( \alpha = 60^\circ \) as compared to other cases. The separation line near rotor tip of maximum stall margin (\( \frac{L}{{C_{\text{R}} }} = 50\% \)) is positioned into the rotor-bleeding port, whereas the position of separation line near rotor tip for the reference rotor-bleeding airflow and maximum stable range extension (\( \alpha = 60^\circ \)) was placed downstream of the rotor-bleeding port that justifies the maximum stall margin at \( \frac{L}{{C_{\text{R}} }} = 50\% \) as compared to other cases.

Fig. 9
figure 9

Relative Mach number contours on 98% span surface of NASA stage 37 at peak efficiency condition with rotor-bleeding airflow: a maximum adiabatic efficiency (\( \frac{W}{{C_{\text{R}} }} = 3\% \)), b maximum total pressure ratio (\( \frac{{\dot{m}_{\text{B}} }}{{\dot{m}_{\text{SC}} }} = 0.7\% \))

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Fig. 10
figure 10

Streamlines on NASA stage 37 blades at near-stall condition with rotor-bleeding airflow: a maximum stall margin (\( \frac{L}{{C_{\text{R}} }} = 50\% \)), b maximum stable range extension (\( \alpha = 60^\circ \))

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4 Conclusion

The performance of a circumferential rotor-bleeding ejector combined with airflow ejection in NASA stage 37, a single-stage transonic axial compressor, was evaluated using 3-D RANS analysis. The numerical results indicated that the circumferential rotor-bleeding ejector with 0.5% airflow ejection significantly delays the near-stall point from 0.9385 normalized mass flow rate for the smooth casing to 0.9250 normalized mass flow rate, with a 0.09% and 0.51% increment in peak adiabatic efficiency and total pressure ratio at peak efficiency condition, respectively. The stall margin and stable range extension are increased by 1.31% and 21.92%, respectively. The equivalent Von-Mises stress on rotor and stator blades are almost constant, only 0.07% and 0.66% increasing as compared to the smooth casing results, whereas the total deformation on rotor tip blades is also decreased, 0.41% reducing as compared to the smooth casing result. From the parametric study results, the maximum stall margin of 12.18% is found at \( \frac{L}{{C_{\text{R}} }} = 50\% \), and this indicates that the bleeding ejector port is located at the midpoint of rotor tip blade chord. The highest total pressure ratio at peak adiabatic efficiency condition of 2.0247 is found at \( \frac{{\dot{m}_{\text{B}} }}{{\dot{m}_{\text{SC}} }} = 0.7\% \), and largest stable range extension of 32.21% is found commonly at \( \alpha = 60^\circ \), and the maximum adiabatic efficiency of 84.12% is found at \( \frac{W}{{C_{\text{R}} }} = 3\% \), smallest of bleeding ejector port. The lowest reduction in total deformation on rotor and stator blades of 1.62% and 1.17%, respectively, illustrates the bester location of bleeding position at the midpoint of rotor tip blade chord.

Based on this parametric study results, further work is needed to optimize the circumferential rotor-bleeding ejector geometry and ejection mass flow rate using the multi-objective optimization technique based on surrogate models to maximize the both aerodynamic and structural performances of the single-stage transonic axial compressor.