Trailing-edge shock loss control with self-sustaining synthetic jet in a supersonic compressor cascade

Yinxin ZHU, Wenqiang PENG, Zhenbing LUO, Qiang LIU, Wei XIE,Pan CHENG, Yan ZHOU

college of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China

KEYWORDS Compressors;Flow control;Mach reflection;Trailing-edge shock;Self-sustaining synthetic jet;Shock waves;Supersonic cascades

Abstract To effectively reduce the loss of strong shock wave at the trailing edge of the supersonic cascade under high backpressure, a shock wave control method based on self-sustaining synthetic jet was proposed.The self-sustaining synthetic jet was applied on the pressure side of the blade with the blow slot and the bleed slot arranged upstream and downstream of the trailing-edge shock,respectively.The flow control mechanism and effects of parameters were investigated by numerical simulation.The results show that the self-sustaining synthetic jet forms an oblique shock wave in the cascade passage which slows down and pressurizes the airflow, and the expansion wave downstream of the blow slot weakens the shock strength which can effectively change the Mach reflection to regular reflection and thus weaken the shock loss.And the suction effect can reduce loss near blade surface.Compared with the baseline cascade, the self-sustaining jet actuator can reduce flow losses by 6.73% with proper location design and vibration of diaphragm.

1.Introduction

Higher requirements for single-stage compressors are entailed to increase the thrust-to-weight ratio which is always the goal throughout the development of aero-engines.1The supersonic cascade can meet the requirements by appropriately organizing the shock waves in its passage.Ramgen Power Systems, Inc.(RPS)2designed a family of shock-in type supersonic cascades by introducing the principles of supersonic inlets.The pressure ratio of the compressor stage was further improved with multiple reflected oblique shock waves formed in the blade passage.The Shock Wave Boundary Layer Interaction (SWBLI)in the supersonic cascade causes shock loss and increases the risk of flow separation.3The structure and location of the shock wave system in the supersonic blade passage have a greater impact on the performance of the cascade with the development of compressors.4–5Reasonable flow control methods can affect the shock wave structure, thus improving the performance of the compressor cascade.Therefore, it is necessary to carry out flow control studies for supersonic cascade.

Considerable researches already have been conducted to control the shock and shock-induced flow separation.6–7Classic control methods include vortex generator, bump structure,suction and blowing.Placing special structures on the surface of the study object can effectively control flow losses.Holden and Babinsky8eliminated the separation induced by a normal shock/turbulent boundary layer interaction in a blowdown wind tunnel with sub-boundary layer vortex generators.Liu et al.9founded that a bump structure can reduce losses induced by the first passage shock and boundary layer separation in a supersonic cascade with high incoming Mach number.The bump structure can reduce losses by 4.6%.

Another control mode is to inject fluid into the mainstream or to remove low-energy fluid from mainstream.Reijnan10experimented on a transonic aspirated rotor and improved the blade performance with the suction control method in cascades containing shock structure.Wang et al.11found that the most effectively location of applying suction control was at the interaction of the shock training leading edge and the suction surface boundary layer in the SAV21 supersonic cascade.Cao et al.12combined the bowed blading with boundary layer suction in a cascade of a high load compressor.The performance of cascade is enhanced by removing the corner separation.Even if the suction coefficient of this method is too high, the great potential is illustrated.Boundary layer blowing is mainly blowing away the low kinetic energy flow near the object surface through high-pressure air flow.13–14Cao et al.15effectively removed the flow separation downstream and reduced the size of the separation zone in a high-loaded cascade with air injection.Microjet is a newer type of blowing method with the advantage of low jet consumption flow and achieves an effect similar to that of a vortex generator.Souverein and Debie`ve16reduced the size of separation bubble induced by SWBLI with air jet vortex generators.Sang et al.17proposed a type of supersonic fluidic oscillator and demonstrated the feasibility of pulsating supersonic jets preliminarily, which is expected to become a new flow control method for supersonic flow field.Boundary layer suction and blowing techniques are simple and effective, but the loss of captured flow and the energy consumption for injecting high-pressure air must be taken into account, respectively.

Flow control methods without air source such as plasma actuation and synthetic jet can avoid the introduction of additional complex devices.18–19Wang et al.20used a NS-DBD(Nanosecond Dielectric Barrier Discharge) plasma actuator on the pressure and suction side of a supersonic cascade.Shock-induced separation on the blade pressure surface was suppressed and the overall total pressure loss of the blade passage was reduced by 7.4%.Sheng et al.21also found that the plasma actuation could reduce shock loss in the supersonic cascade.Ma et al.22investigated the control effect of a blade end slot which induced a jet flow into the corner region under the pressure difference between pressure and suction sides.They reduced the total pressure loss by 9.7%by retraining corner separation.McCormick23placed a porous surface with common plenum in the region of the shock impingement.Experimental study by McCormick revealed a significant reduction of the shock-induced separation.Pasquariello et al.24replaced the porous surface and cavity by a duct to have better control on the effect of the precise location of suction and blowing.This control method extracts energy from the mainstream.Yan et al.25called this structure as the secondary recirculation jet and found that the potions of the bleed and suction slots had a great impact on the flow control efficiency.Du et al.26reduced the size of separation bubble with the secondary recirculation jets but the total pressure loss was intensified with this structure.Self-sustaining Dual synthetic Jet(SDJ)is a new type of supersonic flow field control technology,presented by Luo et al.27of the National University of Defense Technology based on the dual synthetic jet actuator.SDJ actuator consists of two cavities which are separated by a diaphragm and are connected to the external flow field by blow and bleed slots.Airflow is driven into the bleed slot and injected into the external flow field from the blow slot driven by the differential pressure between the two slots.The diaphragm vibrates at a specific frequency to exert energy on the fluid in the cavities.Liu et al.28found that the strength of the original shock induced by the step leading edge can be reduced effectively with SDJ.The SDJ actuator can be simplified to Self-sustaining Synthetic Ket(SSJ)actuator in order to consider the effects of a single cavity, as shown in Fig.1.

At present,most of the studies on flow control in compressor are continued with transonic and low Mach number supersonic cascade.And most control schemes improve the performance of cascades by improving the separation of blade surfaces and corner areas, focusing more on the improvement of flow field quality near the boundary layer.A new control method based on self-sustaining synthetic jet is proposed in this paper.Three-dimensional simulations in the supersonic cascade with strong trailing-edge shock were conducted to control shock loss with SSJ.Section 2 introduces the supersonic cascade model along with the arrangement of the SSJ and numerical method.Section 3 analyzes the control mechanism of the strong trailing-edge shock in the cascade.Sections 3.1 and 3.2 analyze the effect of actuator structure on the flow field.The effects of vibration of the diaphragm are not considered in these sections.SSJ without vibration of the diaphragm refers to Self-sustaining Jet(SJ).Section 3.1 compares the flow field characteristics with and without SJ control.To decouple the effect of the blow and bleed slots of SSJ actuator, Section 3.2 compares the control effects of Self-sustaining Jet(SJ), injection (J) and Suction (S).According to the results of Section 3.2, the actuator structure is improved and the effects of vibration of the diaphragm are considered in Section 3.3.

2.Physical model and numerical setup

2.1.Cascade model description and self-sustaining synthetic jet arrangement

Self-sustaining Synthetic Jet (SSJ) was carried out in a linear supersonic compressor cascade in this study.The SCM-1.75 supersonic cascade was designed with high inlet Mach number by Institute of Engineering Thermophysics, Chinese Academy of Sciences.29The key geometric and aerodynamic parameters of the SCM-1.75 cascade are shown in Table 1.

Fig.1 Sketch of operational principle of self-sustaining synthetic jet actuator.

Table 1 Key geometric and aerodynamic parameters of SCM-1.75 cascade.

The control scheme based on SSJ is shown in Fig.2.The SSJ actuator is simplified using a structure similar to Pasquariello’s24to simulate the zero-net-mass-flux characteristic of SSJ, as shown in Fig.2.The height of the cavity and width of the slots of the actuator are 1 mm.A normal shock marked in orange forms near the trailing edge and interacts with the pressure side of the upper blade.The blow and bleed slots of SSJ actuator were arranged on the pressure side of the blade.As mentioned in the introduction, the operation of SSJ depends on the pressure difference.So, they were located upstream and downstream of the Trailing-Edge shock (TE shock),respectively.The airflow from the blow slot will induce an oblique shock which is parallel to the reflected shocks in the blade passage.If the blow slot is close to the TE shock, the oblique shock cannot decelerate the airflow for the entire passage upstream of the TE shock.And after passing through the oblique shock wave, the airflow will accelerate again.If the blow slot is too far away,the deceleration effect of the oblique shock is weakened.Meanwhile, pressure on the blade surface rises due to the TE shock.Therefore, the distance between the two slots should be far enough to ensure that there is enough pressure difference.To balance jet strength and suction effect, the suction slot was on the leeward side of the separation bubble.So, the blow and bleed slots of the actuator were arranged at about 53% and 65% of chord length initially, as can be seen in Section 3.1 and 3.2.

2.2.Numerical simulation methods

The 3D Reynolds-averaged Navier-Stokes equations were used to simulate the flow field of the SCM-1.75 cascade.The density-based double precision solver of FLUENT was employed with the Roe-FDS flux type.The fluid was assumed to be an ideal gas.

The computational domain was structured by the software Pointwise.Upstream and downstream extension distance of the blade edges were one chord.‘‘O-type”mesh was created around the blade and other domains were ‘‘H-type”.The first mesh height of the blade surface was set to 1 × 10-6m to ensure that the values of y+are less than 1.The computational domains are shown in Fig.3.The pressure far field boundary was set on the inlet boundary with total pressure being given as 380 kPa and total temperature 320 K which were the same as the experimental measurements.29And the inflow angle was set to be 70.5°.The pressure outlet was on the outlet boundary.And the pressure ratio was 2.5.The blade surface was assumed to be adiabatic non-slip wall.The upper and lower boundary were set as translational periodic boundary.The computational domains would be duplicated in the following sections to facilitate the observation of the flow field.In order to reveal the control mechanism of SSJ on the trailing-edge shock on the blade surface, the complicated influence of the threedimensional effect near the end walls was ignored.30The spanwise width of the domains was 20 mm.And borders on the left and right sides were set as the translational periodic boundary.The flow structures would be complex near the leading and trailing edges of the blade, the blow and bleed slots of the SSJ.And the details of the mesh are shown from the front view in Fig.4.

To choose the most suitable turbulence model,three different models were used to predict the distribution of isentropic Mach number on the blade surface of the baseline cascade,which were the Spalart-Allmaras (SA) model, the standard k-ε model and the SST k-ω model.The isentropic Mach number is defined as

where pinand plocaldenote the inlet and local static pressure respectively.The cross-sections of pressure measurement are 10%of the axial chord length from the inlet and outlet boundary.A second-order upwind TVD scheme was applied for the spatial discretization.In Fig.5,all three models could obtain a distribution law of isentropic Mach number on the blade surface consistent with the experimental measurements.29However, the SA model overestimated pressure step and the standard k-ε model predicted higher turbulent kinetic energy of the boundary layer resulting in higher resistance to separation.The SST k-ω turbulence model was adopted.The grid independence was checked with coarse,medium and fine grids consisting of 3.58, 4.93 and 6.77 million cells in the baseline respectively.And the total pressure loss coefficient ω calculated by the fine grid was used as benchmark.The total pressure loss coefficient is defined as

3.Results and discussion

3.1.Flow field characteristics

Fig.2 Schematic diagram of cascade with self-sustaining synthetic jet.

Fig.3 Computational domains of cascade.

Fig.4 Details of grid around blade leading edge, trailing edge,and slots of actuator.

In this section, a cross-comparison between uncontrolled and controlled cases is provided.To analyze the effect of actuator structure on the flow field, the effects of vibration of the diaphragm are not considered in this section.SSJ without vibration of the diaphragm refers to Self-sustaining Jet (SJ).Case 0 refers to the uncontrolled baseline case and Case SJ refers to the case with the control of self-sustaining jet.The parameters of Case SJ have been shown in Section 2.1.The comparison of the flow fields with and without control is given in Fig.6, where contours of streamwise density gradient within the x-y mid-plane are shown.The definition of streamwise density gradient is

Fig.5 Isentropic Mach number distribution on blade surfaces with different turbulence models.

Table 2 Loss coefficient with different grids.

where ρ is the local density and α is the incoming flow angle.The structures of waves can be easily identified through contours where compression waves are red and expansion waves are blue.The Leading-Edge shock wave(LE shock)is reflected on the suction side and multiple reflected shock waves are formed in the passage decelerating and pressurizing incoming flow.Then the boundary layer on the pressure side exhibits a transonic separation induced by Trailing-Edge shock (TE shock).The middle of TE shock is a Mach stem and two ends split into an oblique shock foot and an approximate-normal shock foot or so called λ-foot (highlighted by the blue circles in Fig.6(a)).Since the flow field before the TE shock is supersonic, the flow field structure in the upstream passage is unchanged after applying the control compared with Figs.6(a) and (c).

Fig.6 Contours of streamwise density gradient in baseline and controlled cascades.

Figs.6 (b) and (d) are localized enlarged plots near the TE shock.The x-axis is the dimensionless chord length.The sonic line (Ma = 1) is superimposed in green.The streamlines near the blade surface are highlighted by fuchsia lines.In the baseline case,the separation point is very close to the oblique shock foot.The flow becomes subsonic when passing the approximate-normal shock.The actuator changes the motion of the flow near the wall.Driven by high pressure difference,gas in the separation bubble is sucked into the actuator and injected at the blow slot as a weak jet.The weak jet is deflected after entering the passage immediately and obstructs the mainstream inducing a shock-expansion system.Suction forcibly changes the direction of the airflow within the boundary layer.A part of streamlines are deflected in reverse near the downstream boundary of the suction slot.Thus, a recirculation region appears, which indicates that the effect of the suction is limited.Unlike findings from Pasquariello et al.24and Du et al.,26the change of shock pattern of the baseline flow field is an identifiable feature for SJ control while the height of the separation bubble is not significantly reduced.As can be seen, with the vertical injection of the actuator, the oblique shock foot of TE shock moves upstream and the approximate-normal shock foot moves downstream slightly.The Mach stem disappears.Hence the crossing shock wave interaction becomes regular reflection.

As reflected shock waves gradually weaken in the blade passage, the Mach number of airflow in the blade passage no longer drops significantly, as shown in Fig.6 and Fig.7.Relatively,the strength of shock-expansion system induced by the actuator is great.The oblique shock attenuates flow velocity and pressurizes airflow in the passage effectively.The expansion wave weakens TE shock.Above are the main reasons why TE shock changes from Mach reflection to regular reflection.

3.2.Decoupling of injection and suction effects

The effect of SJ can be simplified to a combination of injection and suction control(The vibration of the diaphragm is ignored in this section).The mass flow rate into the actuator is affected by parameters such as location,angle and local pressure of the blow and bleed slots, which affect both suction and injection control effect.When the mass flow rate through the SJ actuator is small,the ability to inhibit separation of suction and the strength of the jet is weak meantime.Effects of injection and suction need to be analyzed urgently.Therefore, the decoupling of the control effects of the SJ in this section will help modify the control configuration for better control effect.

Fig.7 Contours of Mach number distribution in baseline and controlled cascades.

Case J and Case S are supplemented in this section, as shown in Fig.8.Case J refers to the case with the control of jet and Case S refers to the case with the control of suction.The locations of slots in Cases J and S are the same as in Case SJ.Pressure inlet and outlet were used to simulate the effect of injection and suction of Cases J and S respectively.The total pressure, static pressure and total temperature in the middle of the actuator in Case SJ were measured to be 158.596 kPa,157.383 kPa and 315 K respectively.So,the pressure and temperature in Cases J and S were set to the same values.

The zero streamwise velocity line in black is used to mark the separation zone.As shown in Fig.9, the bleed slot can effectively delay the start of separation and reduce the size of separation bubble in Case S compared with Cases 0 and J.Therefore, the oblique shock foot moves downstream so the height of λ-foot drops and the Mach stem moves towards the pressure side of the blade.The location of the oblique shock foot of the lower blade remains unchanged and is at the trailing edge of the lower blade.Accordingly, the height of λ-foot of the lower blade is raised and the length of the Mach stem is almost unchanged.While the injection effect has opposite influence and moves the separation bubble forward,the shock patterns of Cases SJ and J are similar as analyzed in the previous section.It can be seen that the bleed slot has a good control effect on separation bubble,while the blow slot has a good control effect on shock pattern.

The shock patterns affect the Mach number distribution in the blade passage in Fig.10.In Case S, due to the backward shift of the shock patterns, the airflow has a longer distance to accelerate.Coupled with the acceleration effect of the bleed slot, the Mach number before the TE shock is higher.The Mach number changes more drastically before and after the Mach reflection.The acceleration effect makes the lowvelocity wake of the blade shorter, while the oblique shock induced by blow slot significantly decelerates the airflow in Cases SJ and J.Although the expansion wave accelerates the airflow slightly, the overall flow speed is lower than that in Cases 0 and S.

Fig.8 Decoupling models comparison.

Fig.11 shows the wall pressure along the pressure side of the blade.A pressure rise occurs near the separation point.The gray lines mark the locations of the blow and bleed slots.Due to the obstructive effect of injection on incoming flow near blade wall in Cases SJ and J, the pressure rise appears from once to twice at the anterior and posterior of the blow slot compared with Cases 0 and S.And the second rise is gentler.In Case S, with the backward shift of the oblique shock foot, the pressure rise appears later than other three cases.Hence the pressure is much lower at the anterior of the bleed slot but the gradient of pressure rise is more intense.Another pressure rise occurs at the posterior edge of the bleed slot.Due to the connectivity effect of the SSJ actuator, the pressures of the blowing and bleed slots are similar.The recirculation zone induced by suction causes the fluid flow near blade wall to deflect sharply so the pressure increases.The peak value of the wall pressure is the lowest in Case J and the highest in Case S.And a more uniform type of blade loading is achieved around the trailing edge with the control of injection and SJ.Overall wall pressure changes more moderately in Cases J and SJ than in other cases.

Fig.12 shows the distribution of Turbulent Kinetic Energy(TKE) near the blade wall.The oblique shock and TE shock are marked with dashed white lines.The TKE of airflow near the blade wall increases sharply after passing through the shock foots of the oblique shock and TE shock in all cases.The deflection of airflow towards the bleed slot strongly enhances TKE near the posterior edge of the bleed slot.However, the near-wall flow acceleration induced by the bleed slot contributes to an overall lower TKE.The TKE downstream of the blow slot increases due to the small recirculation zone in this region.Then airflow with high TKE is converted downstream and bulges the shear layer slightly.In Case SJ and Case J, a slight rise of TKE appears with injection.After passing through TE shock, the TKE increases further.

Fig.9 Streamline density gradient contours with different control configurations.

Fig.10 Mach number contours with different control configurations.

Fig.11 Wall pressure distribution on pressure side of blade.

The total pressure loss coefficient ω is calculated using Eq.(2) and shown in Table 3.The changes of mass flow rate in the mainstream in Cases J and S by injection and suction are ignored.The parameter mris the ratio of mass flow rate in the actuator of each configuration to the mainstream.This parameter represents the strength of the suction or injection with slight differences.And the control effect is evaluated by the reduction ratio of ω compared with uncontrolled Case 0.The control effect Δω is defined as

where ωcontroland ω0are the total pressure loss coefficients of the controlled and baseline cases respectively.All configurations have positive control effect.The injection control has excellent effect even though mrin Case J is small.However,the effect of suction is very limited.

The distributions of loss coefficient are shown in Fig.13 and Fig.14.Note that the two sets of contours use different legends in order to observe the loss status in the passage and near the blade wall.Downstream of the Mach stem in Cases 0 and S is a high loss area in Fig.13.Even the losses of some areas have increased in Case S compared with the uncontrolled case.In Cases SJ and J,the loss status within the blade passage is effectively improved due to the disappearance of the Mach stem.

Fig.12 Turbulent kinetic energy contours with different control configurations.

Suction effect mitigates the loss near pressure side of the blade in Fig.14(d).This effect passes all the way to the trailing edge.This is also the reason why the Mach number near the trailing edge has been increased in Fig.10 (d).The high loss areas near the pressure side are thicker in Cases SJ and J compared to the uncontrolled situation,because the obstruction of airflow near the blade surface by injection thickens the high turbulent kinetic energy region.It can be found that the injection can effectively reduce the loss of the passage mainstream,while the suction can reduce the loss of the boundary layer.Since the proportion of mainstream losses is greater, the total pressure loss reduction controlled by SSJ is mainly obtained by the decrease of the trailing-edge shock loss.

Thus, the control effect by SJ is mainly obtained by the injection from the blow slot.Meanwhile, the better control effect is achieved by the injection in Case J.It can be inferred that a lightweight, small, low-energy SSJ actuator can have a jet-like effect without additional air source.And actuator still has potential by improving its structure as can be seen in Section 3.3.

Fig.13 Loss coefficient contours in blade passage.

Fig.14 Loss coefficient contours near blade surface.

Furthermore, shock wave/boundary layer interaction will cause local high thermal load on the blade surface, resulting in local fatigue and ablation and thus threatening compressor safety.The wall temperature is used to monitor the distribution of heat flux on the blade surface, as shown in Fig.15.Near the blow and bleed slots, the local temperature rises because of the complex flow and the high turbulent kinetic energy.The peak values of the wall temperature in Cases 0 and SJ are similar.It can be inferred that the self-sustaining synthetic jet control method will not produce a local heat concentration compared with suction control.

3.3.Effects of diaphragm vibration and an improved configuration

In this section, the effects of vibration are considered and the actuator structure is improved.Case SJ refers to the case with the control of self-sustaining jet consistent with the previous sections.Case SSJ refers to the case with the control of selfsustaining synthetic jet, which can be understood as the Case SJ with diaphragm vibration.The structure parameters of SSJ are the same as those of SJ.As shown in Section 3.2,the injection effect is the key to improving the performance of the cascade.Thus, the control effect depends on a reasonable injection strength.And the injection strength increases as the pressure difference between slots increases, which can be achieved by adjusting the location of the bleed slot.Case SJ-IM refers to the case with an improved structure of SJ.The location of bleed slot in Case SJ-IM moves backwards

Fig.15 Wall temperature distribution on pressure side of blade.to 71%chord length for greater pressure difference.And Case SSJ-IM refers to the case which has the same structure as Case SJ-IM with diaphragm vibration.

To simulate the zero-net-mass-flux characteristic SSJ, the same tubular structure as Case SJ was used in Case SJ-IM.In Cases SSJ and SSJ-IM,the tubular structure was truncated into two parts in the middle, as shown in Fig.16.The spacing between cross-sections is 1 mm.The SSJ actuator is based on the (dual) synthetic jet actuator.Velocity inlet is commonly used in many studies on (dual) synthetic jet(s) in noncompressible flow field.31–33In compressible flow field, especially supersonic flow fields, pressure inlet is more suitable.34Thus, pressure inlet boundary condition was utilized to simulate the diaphragm in SSJ actuator.The mass flow rates of the two boundaries were monitored.The difference of mass flow rates was kept not greater than 2 × 10-4kg/s so that the boundary settings could simulate zero-net-mass-flux characteristic as much as possible.

Fig.16 Schematic diagram of SJ-IM and SSJ-IM model.

Table 4 Loss coefficients of two control configurations with and without diaphragm vibration.

Fig.17 Time-average loss coefficient contours of improved Case SSJ-IM.

Total pressure of the two boundaries was set to a value changing periodically to simulate the diaphragm vibration,which can be defined as

Table 4 gives the loss coefficients of four control configurations.The adjustment of blow and bleed slot locations improves the control capability in Cases SJ-IM and SSJ-IM.Meanwhile, the injection strength increases with the influence of diaphragm on cavity pressure.Thus, combined with the analysis in Section 3.2, the control effect is further enhanced.The control effect is significantly improved, as shown in Fig.17, due to the adjustment of actuator structure and the diaphragm vibration.Compared with Fig.13 (b), the losses are reduced in the area between the oblique shock induced by injection and the TE shock.The downstream flow field of TE shock has a significant reduction in losses.

4.Conclusions and prospect

In this study, the control effect on aerodynamic performance of a supersonic compressor cascade with high inlet Mach number is investigated using self-sustaining synthetic jet.The decoupling of the self-sustaining synthetic jet is conducted to evaluate the influence of the blow and bleed slots and diaphragm vibration.According to the analysis of threedimensional numerical data, the main conclusions can be drawn as follows:

(1) The self-sustaining synthetic jet can significantly change the pattern of trailing-edge shock from Mach reflection to regular reflection, which mainly relies on the effects of injection.Flow parameter changes before and after the shock wave are more moderate.The self-sustaining synthetic jet actuator can reduce total pressure loss coefficient by 6.73% with proper slot location design and vibration of diaphragm.

(2) The control effect of the blow and bleed slots of selfsustaining synthetic jet actuator on the total pressure loss in supersonic cascade is different.The injection of the self-sustaining synthetic jet can effectively reduce the losses in the passage, while the suction can reduce the losses of the boundary layer.Since the proportion of mainstream losses is greater, the control effect of self-sustaining synthetic jet is mainly obtained by the decrease of the trailing-edge shock loss.

(3) The pressure difference can be further increased by adjusting the locations of the two slots to increase the injection strength and thus improve the control effect.

(4) The self-sustaining synthetic jet control can avoid producing a local heat concentration ensuring the safety of the compressor compared with suction and blowing control.

A systematic study on the geometric parameters of the selfsustaining synthetic jet is not conducted in this paper.In the next work, the optimal geometric parameters of the selfsustaining synthetic jet actuator cavity will be explored.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (No.52075538), the National Science and Technology Major Project, China (No.J2019-II-0016-0037), the Natural Science Foundation of Hunan Province,China (No.2020 JJ2030) and the Foundation of National University of Defense Technology, China (No.ZK-22-30).

Particular thanks are given to the Institute of Engineering Thermophysics, Chinese Academy of Sciences for the offer of the geometry of the compressor cascade, and to Yongzhen LIU, Xiaoxiao ZHOU and Qiangren XU for their technical guidance on numerical calculation work.