Numerical study on the stability enhancement mechanism of self-recirculating casing treatment in a counter-rotating axial-flow compressor

ABSTRACT Counter-rotating axial-flow compressor (CRAC) is a promising technology to increase the thrust-to-weight ratio of aero-engines. Self-recirculating casing treatment (SRCT) is selected to explore its stability enhancement capacity in the CRAC. In the present work, an effective SRCT is designed by the design of experiment method, and the stability improvement potential of the SRCT and its mechanism of stall margin enhancement are studied. Results show that the optimal SRCT scheme increases the stall margin by about 7.73% and without remarkable peak efficiency loss, and the overall performance of the compressor is enhanced at the near-stall condition. The SRCT improves the quality of the tip flow field by sucking out low-energy fluid and restrains the spillage of the tip leakage flow by the jet effect from the injection port. In addition, the SRCT also reduces the unsteady interference between the adjacent rotors. The effect of speed ratio on the effectiveness of SRCT is further investigated, and results indicate that the SRCT increases the stall margin of the CRAC by about 4.13–5.80% at some off-design speed ratio conditions. The speed ratio affects the first stall stage of the CRAC, thus affecting the effectiveness of the SRCT.


Introduction
Counter-rotating axial compressor (CRAC) is considered as a promising technology for future aero-engines with a higher thrust-to-weight ratio. Due to the cancelation of the stator between the two adjacent counter-rotating rotors, the stage load of the compressor is improved and both axial-length and weight are shrunken greatly. Besides, the CRAC has a higher pressure ratio and efficiency than conventional compressors if the internal flow is well organized (Nouri et al., 2013;Wang et al., 2013). Unfortunately, the increase of relative velocity aggravates the tip leakage flow (TLF) and flow separation when the loading of the rear rotor increases in the CRACs (Toge et al., 2017). In addition, the counter-rotating effect makes the rear rotor undertake higher aerodynamic load in CRACs, thus the rear rotor usually stalls first (Chen et al., 2019). The speed ratio of the two adjacent rotors is a particular parameter of CRACs and it has a significant influence on the overall performance of the CRAC (Gao et al., 2012). With the front rotor maintaining the design speed, increasing the rear rotor speed can improve the performance of the rear rotor and whole stage (Mistry et al., 2014;Zhang et al., 2021). The aerodynamic CONTACT Limin Gao gaolm@nwpu.edu.cn problems are more complicated in CRACs, therefore, the stall margin of CRACs is generally more limited. The current research in the CRACs mainly focused on the internal flow mechanism, and there is limited research on flow control technology. Knapke et al. (2013) showed that the efficiency of the CRAC is improved by 4.2% with the boundary layer suction. Until 2016, the effect of the casing boundary layer suction (Shi et al., 2015) was investigated experimentally on the CRAC test rig in Northwestern Polytechnical University and found the performance of the CRAC is enhanced by controlling the tip flow, but the optimal suction strategy should be changed with the rotation speed. The casing aspirated in a CRAC conducted by  indicated that the stall margin is improved by about 6.0% at the optimal suction position, but the compressor efficiency dropped by 0.63%.
As a passive control technology, casing treatment has been widely studied in the conventional compressors, researchers also proposed many kinds of casing treatments. For instance, Brandstetter et al. (2016) experimentally investigated the effect of non-axisymmetric casing treatment in a 1.5 stage compressor and results showed that casing treatment reduces the tip blockage, moreover, the stall initiation mode of the compressor is also changed. Mehta et al. (2015) studied the effect of the aspect ratio of circumferential groove on the effectiveness of casing treatment and stated that there is an optimal aspect ratio to achieve the desired stall margin. Parametric studies on the foamed metal casing treatment (Sun et al., 2021) showed that the improvement of stall margin is also accompanied by the efficiency penalty, and the unloading of tip load is the reason for the stall margin improvement. However, Zhou et al. (2017) suggested that the mechanism for the improvement of stall margin is the bleeding and injecting effect of the flow recirculation in the slot weakens the tip blockage. Zhu and Yang (2020) investigated the slots-groove coupled casing treatment on the compressor performance and indicated that the coupled casing treatment harms the peak efficiency. Day (2016) indicated that the stall margin improvement and the efficiency penalty caused by casing treatment are almost linear. However, research on casing treatment in the CRACs has rarely been reported in recent years. Pundhir et al. (1990) first explored the effectiveness of casing treatment in a CRAC and results showed that casing treatment dramatically increases the stall margin and efficiency of the CRAC, but the speed ratio affects its effectiveness. Mao et al. ( , 2020 investigated the circumferential grooves casing treatment in a CRAC and indicated that the stall margin is enhanced by about 5.0%. In addition, the grooves casing treatment improves the stall margin of CRAC and the pressure rise is increased by 4.0%, but the effect of grooves casing treatment on the peak efficiency is not mentioned (Heinrich et al., 2019(Heinrich et al., , 2020. Self-recirculating casing treatment (SRCT) as a novel casing treatment, which not only enhances the stall margin of the compressors but also decreases the efficiency loss caused by the slot/groove casing treatment. Hathaway (2002) pioneered the SRCT in an axial compressor and demonstrated it can broaden the stall margin and almost without efficiency loss. Experimental studies Strazisar et al., 2004) further verified that the SRCT can significantly improve stall margin and reduce the efficiency loss. Studies conducted by Wang et al. (2016) and Kumar (2021) manifested that the SRCT can even increase the peak efficiency at both the design speed and off-design speed.
In conclusion, SRCT has shown great stability enhan cement potential in the conventional compressors, however, there is still a lack of reports on the research of SRCT in the CRACs. Therefore, it is worthwhile exploring the stability improvement potential of SRCT and its mechanism in the CRACs. CFD is an effective tool to explore the flow mechanism in turbomachinery (Akbarian et al., 2018;Li et al., 2020). Thus, a twostage CRAC was selected to investigate the stability enhancement mechanism of SRCT through numerical simulations. Based on the CRAC, the paper first completed the design of an efficient SRCT structure by the DOE method, and then the effect of the injection port position of SRCT and the corresponding stabilization mechanism is investigated. Finally, the influence of the speed ratio on the effectiveness of the SRCT is further explored.

Investigated object
A two-stage counter-rotating axial flow compressor (CRAC) is selected to perform the numerical investigation (Shi et al., 2015;Wang et al., 2015), as shown in Figure 1. The CRAC consists of four-blade rows, i.e. inlet guide vane (IGV, 22 blades), a front rotor (R1, 19 blades), a counter-rotating rotor (R2, 20 blades), and an outlet guide vane (OGV, 32 blades). The two rotors are driven by two motors respectively, which enables the two rotors to operate at different speeds. The compressor produces a total pressure ratio of about 1.22 at the design mass flow rate of 6.4 kg/s. The main parameters of the two rotors are listed in Table 1.

Numerical methods
The computational domain grids were generated by NUMECA/AUTOGRID 5, as shown in Figure 2. The inlet and outlet both adopt the H-type grid, and the tip clearance domain employs a butterfly grid with 25 nodes in the radial direction. The near-wall grids were refined and the thickness of the first layer was set to 5 μm to ensure y + is less than 2.0. In Figure 3, the grid independence of the single passage computational domain shows that when the number of grids exceeds 1.46 million, the total pressure ratio and adiabatic efficiency become insensitive to the grid density. To reduce the cost of numerical simulations, the 1.68 million scheme was selected for the present study. The nodes of grids in axial, radial and circumferential directions are 257 × 81 × 49, and the number of grids in R1 and R2 is about 8.2 and 8.6 million, respectively.
The grids of SRCT were generated by NUMECA/IGG and all grids adopt the H-type grid. As shown in Figure 2, two thin blocks were added at the top of the blade passage to transfer the data accurately. The interface between the two thin blocks was set as the R-S interface condition. The bottom of the lower thin block grid was connected  to the blade passage in the full non-matching connection (FNMB), and the top of the upper thin block was also connected with the bottom of the SRCT in the FNMB. The grid number of SRCT and two thin blocks is about 0.28 million, therefore, the grids of the whole computational domain are about 1.88 million. The commercial software NUMECA/FINE was used to perform the numerical simulations. The threedimensional (3D) Reynolds-averaged Navier-Stokes equations were discretized with the cell-centered finite volume method. The Spalart-Allmaras turbulence model was used to close the discrete equations in the present work due to its successful applications and good performances in turbomachinery (Da et al., 2020;Elfarra, 2019;Wang et al., 2021;Lu et al., 2018). The fourth-order Runge-Kuta method was used for the time discretization. The implicit residual smoothing method and multi-grid technology were adopted to accelerate the convergence process. The inlet direction was assumed to be axial. The total pressure (101,325 Pa) and total temperature (288.15 K) were applied in the inlet. The average static pressure was imposed at the outlet. The adiabatic and non-slip boundary conditions were adopted on the solid walls. The circumferential conservation method was applied to deal with the data transfer in the interface between the two blade rows in the steady simulations. The flow field parameters and performance of the CRAC at different working conditions are obtained by changing the average static pressure of the outlet. In the unsteady simulations, the transfer mode of interface parameters between two blade rows was the slip interface method. The implicit dual-time stepping technique was used in the unsteady simulations. The physical time was specified as 80 steps per blade passage and the pseudo time iteration was set to 20. The steady calculations results were taken as the initial field in the unsteady simulations. Figure 4 presents the overall performance comparison of the CRAC with solid casing (SC) between the numerical calculations (Cal) and experiments (Exp). The comparison shows that the total pressure ratio and adiabatic efficiency obtained by numerical calculations are in good agreement with the experimental results (Wang et al., 2015).

Design of SRCT
The previous study has demonstrated that R2 is the first stall stage of the CRAC at the design speed ratio conditions (Gao et al., 2012). Thus, only R2 is configured with the SRCT in the present work. Figure 5 shows the meridional design diagram of the SRCT, and Table 2 lists the main design parameters. SRCT can utilize the pressure difference between the suction port and the injection port to circulate the low-energy fluid near the suction port to the upstream and reject it into the main flow at a high velocity. The aerodynamic loss in the SRCT is one of the important sources of efficiency penalty, and the efficiency loss can be reduced by optimizing the SRCT structure. To obtain an efficient SRCT structure, the key geometric parameters of SRCT were first studied in the meridional design. The new axial momentum injected by the injector is the main driving force to improve the tip flow field. In order to reduce the flow loss of circulating airflow and obtain higher axial momentum, the suction section adopts a constant cross-section design. The bridge employs a contraction design. The injector design is based on the Coanda effect for its wall-attachment effect .

Parametric study of independent SRCT
There are many design parameters of SRCT, to obtain the key design parameters, the parametric studies of independent SRCT from meridional design to 3D design were first carried out. In the meridional design of independent SRCT, the main geometric parameters include the total height (H) of the SRCT, the air injection angle of the injector (α), the area ratio (S r ) of the bridge inlet (h 2 ) and outlet (h 1 ). The total pressure was applied at the suction port, and the calculation results of six different pressure differences were obtained by changing the static pressure at the injection port. Two objective functions, which are the total pressure recovery coefficient (ζ , which is defined as the total pressure ratio of the suction port and injection port) and axial velocity (V z , the axial component of the velocity at the injection port), were selected to evaluate the effect of each parameter on the aerodynamic performance of the SRCT. That is because ζ can represent the flow loss in the bridge and V z can denote the kinetic energy of the jet. Figure 6 shows the effect of each parameter on ζ (p represents the static pressure ratio of

Symbols
Parameter description Injection position, which is the distance between the injection port and LE of R2 L 2 Suction position, which is the distance between the suction port and LE of R2 C cr Ratio of circumferential coverage of SRCT to single blade passage T ar Throat area ratio, which is equal to A 2 : A 1 S r Shrinkage ratio of bridge, which is equal to h 2 : h 1 the suction port and injection port). It can be noted that ζ declines with the increase of S r and H, and the α has little effect on ζ .
In order to get the key parameters, the geometric parameters of SRCT were taken as experimental factors and analyzed by the design of experiment (DOE) method. The upper and lower values of each parameter are set as follows: H∈ [6 mm, 10 mm], S r ∈ [2, 4], α∈ [5°, 10°]. The main effect graph was selected to analyze the influence intensity of each factor on the objective functions. In the main effect diagram, a positive slope means that the factor is positively correlated with the objective function, whereas the negative slope is the opposite. Figures 7 and 8 illustrate the effect rule and strength of parameters on the two objective functions at two conditions with different driving pressure (i.e. the pressure difference between the suction port and the injection port), which can be summarized as follows. S r has a positive effect on ζ and V z . α has a positive effect on ζ and harms V z , but the effect is weak. H harms ζ and has a positive effect on V z . As the driving pressure difference    increases, the main effect of each factor rises. To reduce the flow loss in the bridge, the bridge should use a uniform cross-section design, that is, S r = h 2 / h 1 = 1.0. However, the selection of α and H needs to be verified with the combination of 3D numerical calculations.
In the independent 3D numerical calculations, three parameters, i.e. the circumferential coverage ratio (C cr ) of SRCT, the throat area ratio (T ar ) between the suction port and the injection port, and the radial deflection angle (γ ) were introduced. Table 2 and Figure 5 present the definition and schematic diagram of each parameter. The DOE method was also applied to find the key design parameter in the 3D design. The objective functions also select ζ and V z . The upper and lower limits of each parameter are α∈ [5°, 10°], H∈ [6 mm, 10 mm], C cr ∈ [40%, 80%], T ar ∈ [2, 4], p∈ [2 kPa, 10 kPa]. It should be emphasized that when the compressor is working at the design point and near-stall point, the pressure difference between the suction port and the injection port is about 4.0 kPa and 8.0 kPa, respectively. Figure 9 presents the effect analysis of different factors on the objective functions with the 3D numerical calculations. The statistical analysis displayed that H harms ζ and has a positive effect on V z , but it is weak. The effect of α is consistent with the numerical calculation results of meridional design. T ar has a positive effect on both ζ and V z , so T ar should be increased as much as possible.
The effect of C cr on ζ and V z is relatively weak. p has a negative effect on ζ and a positive effect on V z .
Based on the above analysis, the key design parameters are found and the influence of each parameter is evaluated by the DOE method. Nevertheless, when the CRAC works under near-stall conditions, the internal physical flow presents a strong 3D unsteady peculiarity. Therefore, the aerodynamic performance obtained by the independent SRCT in numerical calculations can only provide general guidance for structure design. To verify the stability improvement potential of SRCT, it is necessary to combine the SRCT with the CRAC. Consequently, a parametric study of SRCT was carried out in the CRAC. The pressure difference between the SRCT inlet and outlet is closely related to the relative position of the injection port and suction port. Moreover, for a fixed bridge length (L), the driving pressure difference between the two ports is also related to the operating condition of the CRAC. Therefore, H, C cr , γ and L 2 are selected in the following parametric study.

Parametric study of SRCT in the CRAC
The parametric study of SRCT combined with the CRAC was carried out at the design speed ratio. The stall margin improvement (SMI) and peak efficiency improvement (PEI) were selected to assess the comprehensive effect of SRCT with different geometric parameters. The SMI and PEI are defined as follows. where π , η and m represent the total pressure ratio, the adiabatic efficiency and the mass flow rate, respectively. The subscript SRCT represents the self-recirculating casing treatment scheme and SC denotes the solid casing scheme. The subscript NSP and DP signifies the near-stall point and the design point, and PEP denotes the peak efficiency point. Figure 10 compares the effect of different geometric parameters on the SMI and PEI. The results indicate that SMI first rises and then reduces with the increase of H, which manifests that there is an optimal value of H that maximizes SMI, but PEI reduces monotonically. SMI increases monotonically as C cr increases, and PEI does the opposite. With the rise of γ , SMI declines and PEI increases. Increasing the γ of SRCT is beneficial to reducing the efficiency loss, but not conducive to improving SMI.
Given that the function of SRCT is to improve the stall margin, the final geometric parameters are selected as follows. The α, C cr , H and γ of the SRCT were selected as 10°, 80%, 8.0 mm and 0°respectively. The selection of  L 2 and L 1 should be based on the internal flow field of the CRAC. The previous studies have indicated that the tip blockage region is located downstream of 25% axial chord length of R2 (C a2 ) from the R2 LE. Since the role of the suction port is to suck out the low-energy fluid, it was determined that L 2 is 25% C a2 . L 1 was changed to investigate the stability enhancement potential of SRCT and the corresponding mechanism under different axial injection positions. Four different axial injection positions were designed in this study and they were named as SRCT_1 (L 1 = 5.0% C a2 ), SRCT_2 (L 1 = 10% C a2 ), SRCT_3 (L 1 = 20% C a2 ), SRCT_4 (L 1 = 40% C a2 ) respectively, and the schematic diagrams of the four SRCT schemes are shown in Figure 11. Figure 12 displays the comparison of the SRCT and SC schemes. The abscissa is the dimensionless result based on the design mass flow rate. It can be found that all SRCT schemes can broaden the stable working range of the CRAC. As the injection position moves forward, the stability enhancement capability of SRCT increases. According to equation (1), the stall margin improvement (SMI) of SRCT schemes improved by 0.88%, 6.29%, 7.41%, and 7.73% respectively. Calculated by equation (2), the peak efficiency dropped by 1.04%, 0.93%, 0.80%, and 0.60% respectively. Although the SRCT has a negative effect on the peak efficiency, the SRCT is more efficient at the NSP conditions. It can be observed from Figure 12 that SRCT significantly improves the total pressure ratio and adiabatic efficiency of the CRAC in the AB range, and the following sections will analyze the reasons. Figure 13 compares the static pressure coefficient (C p , which is defined as the ratio of local static pressure to the inlet total pressure) of the two rotors on the 99% blade span at the near-stall point (NSP) condition for the SC and SRCT schemes. Figure 13 manifests that the SRCT mainly affects the tip load of R2. To illustrate the change of the static pressure gradient of the blade tip in the pitch direction, Figure 14 represents the C p contours on the 99% blade span of R2.

Effect of SRCT on the tip load
In Figure 13(b), the SRCT significantly increases the pressure difference between the two sides of the blade near the LE of R2 (zone A), which indicates that the aerodynamic load near the LE of R2 raises. The primary factor affecting the pressure in this zone is the main tip leakage flow of the rear rotor. In Figure 14, it can be found that there is a stagnation point near the LE of R2 in the SRCT schemes and it is formed when the jet from the injection port impinges on the LE of the blade. Therefore, the formation of the stagnation point leads to the increase of static pressure on the pressure side in zone A. In zone B, the variation tendency of C p is the same at four SRCT schemes. The C p of both the pressure side and suction side is lower than that of the SC scheme, while the C p of the suction side is opposite. From Figure 14, it can be observed that there is a high-pressure zone (the black dashed rectangular N zone) near the LE of the pressure in the SC scheme. In the SRCT schemes, the trajectory of the TLF is closer to the blade suction side, which induces a low-pressure zone on the suction side. Besides, the joint effect of the jet produced by the injection port and the suction of the suction port accelerates the airflow velocity on the pressure side. Thus, the C p of both the pressure side and suction side declines in region B.
In Figure 14, the static pressure between the front and rear suction port of the SRCT is affected by the suction effect. A high-pressure spot appears on the pressure side of R2 (region M marked by the red dotted circle), which causes an increase in static pressure on the pressure side of R2 (region C in Figure 13(b)). Similarly, the static pressure on the suction side of the blade in the SRCT schemes reduces the blockage of the low-energy fluid due to the suction effect of the suction port, thus the static pressure increases. In region D, the static pressure of the suction side in the SRCT schemes are higher than the SC scheme.  The reason is that SRCT changes the trajectory of the TLF. From Figure 14, it can be found that the trajectory of the TLF (the black dotted line) is close to the suction side and the TLF successfully flows downstream. Figures 13  and 14 show that the pressure of the R1 blade tip with SRCT is unchanged. Therefore, the following analysis will be focused on R2. Figure 15 shows the distributions of the radial average value of the axial velocity (V z ) and the tangential velocity (V y ) of TLF at the split face of the tip clearance. When V z is negative, it indicates that the backflow occurs. The greater the absolute value, the stronger the backflow. Figure 15(a) depicts that the SRCT significantly changes the flow field of the R2 blade tip, and the backflow is suppressed. In Figure 15(b), the V y of the TLF increases across the whole chord, which denotes that the tangential momentum of the TLF is improved and the ability to resist strong back pressure is enhanced. Moreover, it can be noted that the variation trend of the TLF in the SC scheme is relatively moderate, while in the SRCT schemes, the V z and V y of TLF both change suddenly at 30% C a2 (region M). Since the suction port is near region M, the suction effect of the suction port has a distinct impact on the TLF.

Effect of SRCT on the TLF
To further illustrate the impact of SRCT on the TLF, Figure 16 depicts the 3D streamlines of the R2 blade tip. For the convenience of the statement, R2 chord length is equally divided into three parts: Front, Middle, and Trail. In the SC scheme, the TLF induces a large range of backflow since the intense reverse pressure gradient and the blade tip blockage, which further results in the leadingedge spillage (purple solid line). In addition, there is an obvious secondary tip leakage flow (S-TLF, purple dotted line), which affects the adjacent blade passages. However, the leading-edge spillage disappears in the four SRCT schemes. The main tip leakage flow (M-TLF, black solid line) of the blade tip is located at the middle of the blade passage and the S-TLF disappeared. It can be inferred that when the injection port is close to the leading edge, although it acts on the TLF more closely, the pressure difference between the injection port and the suction port declines, so the jet effect is weak and it fails to fully mix with the main flow. In the front range, the jet of the injection port has a stronger restraint effect on the TLF. In the trail range, the jet effect becomes weaker and its inhibition effect on the TLF is also weakened.
In Figure 16, it is difficult to distinguish the influence of different SRCT schemes on TLF streamline. To further compare the effect of the position of the injection port on the stabilization potential of SRCT, Figure 17 compares the airflow velocity of the injection port and suction port (which is calculated by the average integral of the mass flow of the plane, the negative value represents the velocity direction). It can be found that the variation trends of the airflow velocity of the two ports in different SRCT schemes are consistent, but the velocity values are different. In Figure 17, the velocity of both the injection port and suction port gradually increases with the forward movement of the injection port. The increase of the airflow velocity of the injection port is beneficial to strengthen the suppression of the leading-edge spillage, and the increase of the airflow velocity of the suction port can improve the tip blockage. Therefore, the SRCT_4 scheme has the greatest stabilization potential.

Unsteady effect of SRCT
Both the upstream wake and potential interactions are the primary mechanisms related to the unsteadiness in turbomachines . Figure 18 shows the distribution of the C p contours on the 99% blade span of the two rotors at different times at the NSP condition. In the SC scheme, the high-pressure spot (purple dashed circles) appears near the leading edge of R2 at all the time. The high-pressure spot is induced by the downstream potential effect periodically propagates to the upstream. In the SRCT_4 scheme, the high-velocity jet generated by the injection port of SRCT suppresses the downstream potential effect, and no high-pressure spot is observed at different times in the SCRT_4 scheme. Therefore, the SRCT weakens the mutual interference between the two adjacent rotors. In addition, the suction and jet coupling effect of SCRT restrains the intensity of TLF, the spillage behavior near the leading edge of R2 disappears, and the trajectory of TLF is closer to the suction side. Compared with the SRCT_4 scheme, the trajectory of TLF in the SC scheme is related to time, while the trajectories of TLF at different times in the SCRT_4 scheme are consistent, which indicates that the SRCT decreases the flow instability of TLF.
The variation of the C p on the 99% blade span of the two rotors at different times in Figure 19 can more clearly reveal the influence of SRCT on the C p of the blade tip in Figure 18. Figure 19(a) depicts that the C p on R1 in the SRCT_4 scheme is lower than those in the SC scheme, especially near the trailing edge (A region). Since the SRCT decreases the potential flow disturbance downstream, the pressure fluctuation intensity near the trailing edge of R1 is reduced in the SRCT_4 scheme. In Figure 19(b), compared with the SC scheme, the C p distributions near the leading-edge of R2 (region B) in the SRCT_4 scheme at different times are consistent, which indicates that the SRCT reduces the variation of the relative position of the two rotors to the unsteady flow and the disturbance of the wake of R1 on the inlet of R2. In addition, the change of the relative position of SRCT and R2 affects the trend of pressure fluctuation downstream of the suction port (region D). The suction effect of the suction port accelerates the airflow upstream of the suction port, as a result, the C p of the upstream of the suction port (region C) is decreased. The SRCT_4 scheme also  reduces the load near the trailing edge of R2 (region E), thereby suppressing the strength of the downstream TLF.
For the purpose of exploring the influence of SRCT on unsteady interference between the two adjacent rotors, the unsteady fluctuation intensity S u was defined to represent the strength of pressure fluctuation on the blade surface, and its definition can be referred to the reference (Gao et al., 2015). The unsteady flow near the blade tip of the SC and SRCT schemes is further illustrated by the S u . Figure 20 shows the distributions of S u near the casing wall at the NSP condition. Compared with the SC scheme, SCRT can significantly reduce the unsteady fluctuation intensity of the tip flow field. The unsteady fluctuation near R2 LE in the SRCT_4 scheme is weaker (solid rectangle C), which is due to the SRCT inhibiting the instability of TLF. Additionally, the range of the fluctuation at the interface (dotted rectangle B) between rotors is shrunk, and the intensity of the fluctuation on the pressure side of R1 (solid triangle A) is also reduced. The SRCT also decreases the dynamic unsteady  interference between the two rotors, and the jet of the injection port wakens the downstream potential effect. However, the intensity of fluctuation below the suction port and the injection port in the SRCT_4 scheme is increased, which is due to the suction effect and the jet increasing the unsteadiness of the local flow field. Figure 21 shows the distribution of inlet flow angle along the blade span of R2 at different times under the NSP conditions. The SRCT significantly reduces the inlet flow angle within the range of 70% ∼ 100% blade span of R2. In the SC scheme, the leading-edge spillage of TLF occurs at the R2 tip at the NSP condition, which induces a sharp increase in the inlet flow angle of the R2 tip region, while in the SRCT_4 scheme, the inlet flow angle decreases and becomes more uniformly distributed in the whole blade span range. From 2/80T→ 40/80T → 60/80T, the inlet flow angle of R2 decreases integrally. In the SC scheme, with the change of the relative position between the two rotors, the wake and the TLF from R1 change accordingly the inflow of R2, which causes the inlet flow angle of R2 varies sharply at different times.
However, in the SRCT_4 scheme, the variations of the inlet flow angle of R2 at different times are consistent, which manifests that the SRCT weakens the unsteady interference effect between the two rotors, and improves the flow near the tip region of R2, so the stall of R2 is delayed.

Effect of speed ratio on SRCT effectiveness
The speed ratio (SR, the ratio of N 1 to N 2 , N 1 is the speed of R1 and N 2 is the speed of R2) is a unique design parameter of the CRACs compared to conventional compressors. Previous studies verified that the speed ratio affects the aerodynamic performance, stage matching and stall characteristics of the whole compressor (Mistry et al., 2013). The above SRCT design was carried out at the design speed ratio, and the results show that the SRCT can achieve considerable stall margin improvement. However, the SRCT may be incapable of achieving stability improvement at the off-design speed ratios. To investigate the stall margin improvement potential of SRCT at the off-design speed ratios, the SRCT_4 scheme with the best stability enhancement capacity was selected to further research the effectiveness of SRCT at different speed ratio conditions.
The research scheme is divided into two parts, one is to fix the N 1 = 8000 rpm and change the N 2 to obtain six speed ratios of 1.43, 1.25, 1.11, 1.0, 0.91, and 0.83 respectively, the other is to fix the N 2 = 8000 rpm and change the N 1 to get six speed ratios of 0.7, 0.8, 0.9, 1.0, 1.1, and 1.2 respectively. It should be noted that in the above two schemes, the speed change step N = 800 rpm. Figure 22 shows the effect of SRCT on the overall performance of the CRAC at different speed ratios. The abscissa is normalized with the maximum choked mass flow rate. The stability enhancement potential of SRCT at off-design speed ratios is evaluated by the improvement of the stall margin of the mass flow rate ( SM), which is defined as follows. Where m SRCT and m SC respectively represent the minimum stable working mass flow rate at the corresponding scheme.

Effect of SRCT on the CRAC performance
As can be observed from Figure 22, the SRCT can only improve the stall margin under partial speed ratio conditions. According to equation (3), the improvement of SM at different speed ratio conditions is tabulated in Table 3, which indicates that the SRCT does not work at all the speed ratio conditions. Figure 23 presents the two-dimensional streamlines and entropy contours on the 99% span of the two rotors at different speed ratios of the SC scheme. Generally, the interface (purple dotted line in Figure 23) formed by the interaction between the TLF and the main flow is pushed out of the passage at the near-stall point, which causes a leading-edge spillage, the stall has occurred (Vo et al., 2008). The development trend of the streamlines shows that the flow field of the blade tip is closely related to the speed ratio. When the N 1 is fixed at 8000 rpm and the SR is not less than 1.11, both interfaces of R1 and R2 are pushed out of the blade passage, which indicates R1 and R2 enter stall simultaneously. When the SR is less than 1.11, the suction effect of R2 improves the tip flow field of R1, then R1 exits the stall and R2 stalls. When the N 2 is fixed at 8000 rpm and the SR is less than 1.1, R2 is the first stall stage due to the reason that only the interface of R2 is pushed out the blade passage. Both R1 and R2 appear leading-edge spillage when the SR exceeds 1.1, which indicates that the two rotors enter the stall simultaneously. Figure 24 shows the two-dimensional streamlines and entropy contours on the 99% span of the two rotors at different speed ratios of SRCT schemes. It can be observed that when the N 1 is fixed at 8000 rpm and the SR is not less than 1.11, the leading-edge spillage of R2 disappears. The blockage induced by the backflow is reduced, and R2 exits the stall state. The backflow intensity of R1 is raised and the interface is pushed out the blade passage, so R1 stalls first. When the SR is less than 1.11, the two rotors are in a stable state, because there is no leadingedge spillage in both R1 and R2. When the N 2 is fixed and the SR is less than 1.1, the leading-edge spillage of R2 vanishes, and the two rotors are in a stable state. With the increase of SR, R1 is subjected to a stronger reverse pressure and the leading-edge spillage caused by the TLF is more serious, then R1 enters from an unstable state.

Effect of SRCT on the tip flow field
The stall margin of the CRAC is determined by both R1 and R2, and the compressor operates safely only when R1 and R2 are both in a stable state. The comparison  between Figures 23 and 24 illustrates that the first stall stage of the CRAC is related to the speed ratio and the SRCT can change the first stall stage by acting on the blade tip flow field. The stall margin of the CRAC can be improved only by configuring SRCT for R2 when R2 stalls first. When R1 is the first stall stage, the flow field of R2 can be improved by SRCT, but it has no effect on the stall margin of the whole stage. This also explains why SRCT failed to decrease the stall mass flow rate at a partial speed ratio.
In Figure 24, it also can be noted that a high-entropy region appears in the SRCT scheme, and the region is located near the injection port. The reason is that there is a velocity difference between the jet and the main flow, and the interaction between the jet and the main flow leads to mixing loss. Compared with Figure 23, the improvement of the flow field in the tip region by the SRCT suction port can reduce the secondary flow loss. Figure 12 presents that the adiabatic efficiency of the SRCT scheme is higher than that of the SC scheme at the near-stall condition, which indicates that the positive effect of SRCT sufficiently compensates for the mixing loss caused by the jet.

Conclusions
In the present work, the stability enhancement potential of self-circulating casing treatment (SRCT) and  the corresponding mechanism in a two-stage counterrotating axial flow compressor (CRAC) has been studied by numerical method. The effect of injection port position on the performance of the CRAC at the design condition and the effectiveness of SRCT at the off-designed speed ratios are also investigated. The stability improvement potential of SRCT is confirmed and the relevant mechanism is revealed. Some main conclusions can be drawn as follows.
(1) Based on the CRAC, an efficient SRCT is designed from meridional design to 3D design by the DOE method. Due to the interaction between different key geometric parameters of the SRCT, the selection of key parameters requires a balance between the stability enhancement potential and the efficiency penalty.
(2) The stabilization potential of SRCT increases with the forward movement of the injection port. At the design speed ratio, the stall margin improvement is improved by about 7.73%, and the adiabatic efficiency is increased by about 0.90% at the near-stall point. At the off-design speed ratios, the stall margin of the CRAC is also enhanced by about 4.13-5.80% with the SRCT. (3) The stability enhancement mechanism of SRCT in the CRAC includes two aspects. One is that the highvelocity jet from the injection port suppresses the leading-edge spillage. The other is that the suction effect of the suction port alleviates the tip blockage. Additionally, the SRCT restrains the unsteady interference between the adjacent rotors. (4) The speed ratio can change the first stall stage of the CRAC, thus affecting the effectiveness of the SRCT. When the SRCT is employed on the top of R2, the stability improvement potential of SRCT can be achieved only when R2 is the first stall stage or R1 and R2 stall simultaneously. Therefore, the effect of speed ratio on the first stall stage should be considered when performing the stall delay technology in CRACs.

Disclosure statement
No potential conflict of interest was reported by the author(s).