5.1 Smooth casing case

This section examines the aerodynamic performance of the compressor in its baseline configuration, termed the smooth casing (SC) case. In Fig. 5, the speedline for the reference stage is presented, showing the mass flow averaged total-to-total polytropic efficiency ( ${\bar \eta _{{\rm{tt}}}}$ ) and the mass flow averaged total pressure ratio ( ${{\bar{\rm \Pi }}_{{\rm{tt}}}}$ ), plotted against the normalised mass flow rate with respect to the design point mass flow rate ( $\dot m/{\dot m_{DP}}$ ), differentiating between stage and rotor.

Figure 5. Speedline showing mass flow averaged total-to-total polytropic efficiency ( ${\bar \eta _{{\rm{tt}}}}$ ) and total pressure ratio ( ${{\bar{\rm \Pi }}_{{\rm{tt}}}}$ ) for the compressor stage and rotor.

For a perfect gas with constant $\gamma $ (air: $\gamma = 1.4$ ), these quantities are defined as:

(1) \begin{align} {\eta _{tt}} = \frac{{\gamma - 1}}{\gamma }{\rm{\;}}\frac{{{\rm{ln}}\left( {{P_{t2}}/{P_{t1}}} \right)}}{{{\rm{ln}}\left( {{T_{t2}}/{T_{t1}}} \right)}},\end{align}

(2) \begin{align} {{\rm{\Pi }}_{tt}} = \frac{{{P_{t2}}}}{{{P_{t1}}}},\end{align}

where ${P_{t1}}$ and ${T_{t1}}$ are the mass–averaged total pressure and temperature at inlet of the rotor-stator stage (station 1 in Fig. 3), and ${P_{t2}}$ and ${T_{t2}}$ are the corresponding values at the stage outlet downstream of the stator (station 2).

At the design point, the stage achieves a total pressure ratio ${{\bar{\rm \Pi }}_{{\rm{tt}}}}$ of 1.0376, a polytropic efficiency ${\bar \eta _{{\rm{tt}}}}$ of 88.67%, and an inlet corrected mass flow ${\dot m_{DP}}$ of 17.922 kg s^-1. The numerical speedline indicates that the compressor operational range is constrained, with numerical stall occurring at approximately 85% of ${\dot m_{DP}}$ , corresponding to ${\dot m_{stall}} \approx 15.1$ kg s^-1. Here, ${\dot m_{stall}}$ is defined as the minimum mass flow rate at which the simulation remains numerically stable. The rotor performance significantly influences the overall stage behaviour, dominated by a double-leakage tip clearance flow [Reference Hopfinger and Gümmer16]. Accordingly, the current study focuses on stator endwall injection to enhance performance rather than expanding the stall margin.

Figure 6 presents normalised spanwise mass-averaged profiles of key aerodynamic parameters within the stator: the stage polytropic efficiency ${\bar \eta _{tt}}$ , the static pressure rise coefficient $\overline {{C_p}} $ (ratio of static to dynamic pressure rise), the total pressure loss coefficient $\bar \omega $ (normalised total pressure loss) and the axial velocity $\overline {{u_z}} $ at both the leading and trailing edges.

Figure 6. Normalised spanwise profiles at design point : (a) stage efficiency ${\bar \eta _{{\rm{tt}}}}$ , (b) stator $\overline {{C_p}} $ , (c) stator $\bar \omega $ , and (d) axial velocities $\overline {{u_z}} $ at the leading edge and trailing edge of the stator vanes.

As observed, the total pressure loss coefficient exhibits a pronounced increase towards the casing, correlating with the efficiency drop and indicating significant aerodynamic losses. This increase is consistent with the abrupt rise in $\overline {{C_p}} $ over 80% span, suggesting a rapid conversion of dynamic pressure into static pressure. The axial velocity profiles at the leading and trailing edge of the stator vanes further reinforce these observations: at midspan, a natural reduction from annulus divergence and density rise is visible. However, the reduction is more pronounced near the casing starting at $H = 0.7$ in the trailing edge profile, contrasting with the less pronounced velocity reduction at the leading edge, beginning at about $H = 0.85$ . This pattern suggests a growth in boundary-layer thickness, influenced by the adverse pressure gradient, leading to the deceleration of fluid particles which can precipitate flow separation. Additionally, the blockage from upstream rotor tip leakage contributes to low-momentum flow, further impacting stator performance. Similar phenomena are observed near the hub, associated with the shrouded cavity interaction, although we omit their detailed discussion since the focus is placed on the near-casing endwall region.

The flow behaviour over the blade surfaces can be examined in Fig. 7. Figure 7(a) shows the blade loading profiles of the isentropic Mach number $M{a_{is}}$ at two critical spanwise locations, mid-span $H = 0.5$ and near the casing $H = 0.9$ . The profiles are plotted against the normalised chordwise location $s$ , which extends from $s = 0$ at the leading edge of the front vane to $s = 1$ at the trailing edge of the rear vane. Figure 7(b) presents the corresponding limiting streamlines and the contours of the friction coefficient ${C_f}$ . A pronounced loading is observed on the front vane compared to the rear vane, suggested by the greater separation between the suction and pressure sides.

Figure 7. (a) Isentropic Mach number profiles at mid-span ( $H = 0.5$ ) and near the casing ( $H = 0.9$ ), plotted against the normalised chordwise location $s$ from the leading edge of the front vane to the trailing edge of the rear vane; (b) Limiting streamlines and friction coefficient ${C_f}$ contours.

This is confirmed by an overall diffusion factor $DF$ of 0.52 considering an equivalent chord length, while the individual values for the front and rear vanes are 0.44 and 0.27, respectively. At $H = 0.5$ , the $M{a_{is}}$ profiles exhibit smooth gradients for both blade profiles and consistently higher loading along the chord compared to $H = 0.9$ . This contrast indicates the influence of three-dimensional effects and endwall interactions. At $H = 0.9$ the rear vane shows a flattening of the $M{a_{is}}$ profile around $s \approx 0.7$ , which signals a reduced diffusion capacity due to flow detachment. The limiting streamlines confirm the near-casing corner separation, consistent with the local decrease in ${C_f}$ . Such complex secondary flow phenomena at the endwall region in tandem configurations have previously been attributed to the interaction between the front and rear vanes [Reference Baojie, Zhang, An, Du and Yu29].

These findings imply that the aerodynamic performance can be further improved, potentially through targeted flow control strategies. The subsequent sections will discuss the implementation of endwall injection as a means to address these identified shortcomings and thereby improve the stator and stage performance.

5.2 Baseline injection case

A baseline injection case to conduct parametric studies on various geometrical variables is considered. The anticipated flow separation zone on the suction side of the RV suggests that locations on the suction side are promising for flow energisation via air injection. Accordingly, the baseline location for air injection has been selected at roughly 40% of the rear vane chord length. Following the definition presented in Section 3, the normalised chordwise location, denoted by $\zeta $ , is set to 1.4. Table 3 outlines the preliminary design space parameters and the values selected for the baseline case.

Table 3. Design space parameters and baseline values; dimensional parameters are expressed with normalised counterparts in %

${R_c}$ normalised by equivalent tandem stator chord ${c_{eq}} = 0.16{\rm{m}}$ ; ${w_c}$ normalised by local spacing at the casing ${s_{casing}} = 0.086{\rm{m}}$ ; ${w_{a,in}}$ and ${w_{a,inter}}$ normalised by average blade thickness ${t_{mean}} = 6{\rm{mm}}$ ; ${T_{t,inj}}$ normalised by mean total temperature of the 3.5-stage reference compressor ${T_{t,mean}} = 294{\rm{K}}$ .

The injection total temperature, ${T_{t,inj}}$ , is set at 294 K, corresponding to the arithmetic mean of the annulus mass-averaged total temperatures at the inlet and outlet of a 3.5-stage reference simulation (288 K and 300 K, respectively). This choice reflects the mean compressor thermal state, consistent with the assumption that the injection source is extracted from later stages [Reference Khaleghi, Teixeira, Tousi and Boroomand30]. Expressed in normalised form, this corresponds to ${T_{t,inj}}/{T_{t,mean}} = 100{\rm{\% }}$ , where ${T_{t,mean}} = 294{\rm{K}}$ . This avoids unrealistically favourable cooling effects, ensuring the injected air temperature is representative of conditions in the injection region. Additionally, a turbulence intensity of 5% was prescribed at the injector inlet, consistent with the compressor inlet boundary condition.

To ensure minimal disturbance to the main flow, initial values are chosen conservatively. In light of prior research [Reference Denton31], minimising injection angles is critical; hence, the adoption of relatively low $\alpha $ values. The initial injection jet is aligned tangentially with the blade at the injection location ( $\beta = 0$ ). To minimise losses in efficiency, it is preferable to keep the injection mass flow rate to a minimum [Reference Evans and Hodson32]. The injection mass flow rate is normalised with respect to the stall mass flow rate ( ${\dot m_{inj}}/{\dot m_{stall}}$ ), where ${\dot m_{stall}}$ is taken from the smooth-casing simulation. The initial widths of the injector are equivalent to the average blade thickness, being specified as 6 mm.

To improve transferability, dimensional injector parameters are reported together with normalised counterparts. The Coandă radius is scaled by the equivalent tandem stator chord ${c_{eq}} = 0.16{\rm{m}}$ , the circumferential slot width ${w_c}$ by the local spacing at the casing ${s_{casing}} = 0.086{\rm{m}}$ , and the injection temperature by the mean total temperature of the 3.5-stage compressor ${T_{t,mean}} = 294{\rm{K}}$ . For the present study, the widths defined at the inlet ${w_{a,in}}$ and at the interface ${w_{a,inter}}$ remain constant and are normalised by the average blade thickness ${t_{mean}} = 6{\rm{mm}}$ . Expressing these values in percentage form highlights their scale relative to the compressor geometry, while absolute values are retained for clarity.

The boundary condition approach for mass flow control in the baseline case, as well as in the subsequent parametric and location studies, follows the ‘equal outlet mass flow’ method described in Section 4, i.e. the outlet mass flow rate is imposed and set to match that of the smooth-casing case at the design point. For completeness, the baseline case was also repeated using the alternative ‘equal inlet mass flow’ strategy from Section 4, which likewise imposes the outlet mass flow rate but determines its value so that the resulting inlet mass flow matches the smooth-casing case. Due to the small injection fraction employed, differences in stage total pressure ratio and efficiency were within $ \pm 0.016{\rm{\% }}$ and $ \pm 0.03{\rm{\% }}$ , respectively. In the subsequent sections, all reported increments were above these thresholds. Additionally, we selected an exit Mach number for the injector ( $Ma \approx 0.35$ ), which falls within the range of the maximum $Ma$ observed in the compressor, to ensure a realistic operational context. This conservative approach avoids inducing high-speed flow phenomena that could compromise flow stability and aerodynamic integrity.

To provide context for the injection conditions, Fig. 8 presents the local flow field for the baseline injection case at the design point showing velocity and total temperature contours. The contours are shown on an axial plane located immediately downstream of the slot exit, cropped to the upper 50% span for clarity. The jet core patch, centred on the high-velocity jet emerging from the slot within the upper 10% span, is indicated in the figure. For comparison, Table 4 reports mass-averaged flow properties for this jet core patch as well as for the whole-pitch average in the same upper-span band, with the smooth-casing whole-pitch average also listed for reference. The high-velocity jet core is clearly distinguishable from the surrounding endwall flow, while the small differences between the smooth-casing and injection whole-pitch averages confirm that the influence is local. In the jet core, total temperature and total pressure are higher than the surrounding flow, while static temperature and pressure are slightly lower, consistent with the higher local Mach number of the injected stream.

Figure 8. Baseline injection case at design point. Velocity magnitude (left) and total temperature (right) on an axial plane just downstream of the slot exit, cropped to the upper 50% span. The dashed box marks the jet core patch (upper 10% span) used in Table 4.

5.3 Injector parametric study

Initially, we examine the isolated impacts of key injection parameters such as the inclination angle ( $\alpha $ ), jet angle ( $\beta $ ), and radius of curvature ( ${R_c}$ ), keeping the normalised chordwise location ( $\zeta $ ) at 1.4. After establishing optimal values for $\alpha $ , $\beta $ and ${R_c}$ , our focus shifts to assess the interaction between the circumferential width ( ${w_c}$ ) and the relative injection mass flow rate with respect to the stall mass flow rate ( ${\dot m_{inj}}/{\dot m_{stall}}$ ), given their combined influence on achieving the desired exit injector Mach number within specified limits. This process refines our baseline towards an enhanced configuration. Finally, we explore variations in $\zeta $ values to determine effective injector locations.

The subsequent figures illustrate the influence of injection parameters on key aerodynamic performance metrics, showing variations ${\rm{\Delta }}$ for increments in stage variables such as polytropic efficiency ${\eta _{tt}}$ and total pressure ratio ${{\rm{\Pi }}_{tt}}$ , which were defined previously in Equations (1) and (2). The percentage variation ${\rm{\Delta }}$ for any performance metric, such as ${\eta _{tt}}$ or ${{\rm{\Pi }}_{tt}}$ , is calculated according to Equation (3) as

(3) \begin{align} {\rm{\Delta }} = \left( {\frac{{{\varphi _{inj}}}}{{{\varphi _{SC}}}} - 1} \right) \times 100{\rm{\% }},\end{align}

where ${\varphi _{inj}}$ represents the value of the variable for the injection case, and ${\varphi _{SC}}$ represents the value for the smooth casing case.

These parameters are evaluated as mass-flow averaged values at the specified stations 1 and 2. The analysis utilises standard definitions, primarily focusing on the aerodynamic impacts without directly accounting for the costs associated with injection. However, the influence of the injected flow is inherently captured in the outlet conditions, affecting the polytropic efficiency and total pressure ratio. This relationship is further explored through the use of the momentum coefficient ${C_u}$ , which correlates with the aerodynamic performance improvements observed. Section 5.9 provides a detailed discussion of injection losses and introduces the alternative total pressure loss coefficient ${\omega _t}$ , which incorporates the contributions of both the main and injected flows.

Table 4. Mass-averaged flow properties in the jet core patch and in the whole pitch of the upper 10% span for the baseline injection and smooth-casing

5.3.1 Influence of inclination angle

Figure 9 shows how the injector slot inclination angle $\alpha $ affects aerodynamic performance; smaller values of $\alpha $ enhance stage efficiency. The trend confirms that a more tangential introduction of flow reduces disturbance to the main passage and promotes smoother integration of the injected jet, consistent with the findings of Cao et al. [Reference Cao, Gao, Zhang, Zhang and Liu3]. In the present study, the lowest tested angle of $\alpha = {2.5^ \circ }$ yielded the largest efficiency and total pressure ratio increments ( ${\rm{\Delta }}{\eta _{tt}} = 0.463{\rm{\% }}$ , ${\rm{\Delta }}{{\rm{\Pi }}_{tt}} = 0.037{\rm{\% }}$ ).

Figure 9. Aerodynamic influence of inclination angle $\alpha $ , schematic inset shows definition of $\alpha $ .

However, from a numerical perspective, angles approaching zero can present challenges in maintaining mesh quality, particularly after the slot curvature, while from a design perspective, very shallow angles may require higher injection pressures to overcome flow resistance. For these reasons, an angle of ${5^ \circ }$ was adopted in the enhanced configuration as a practical compromise, ensuring sufficient mesh quality and realistic injection requirements while preserving the aerodynamic benefit of a low inclination. Overall, the results suggest that for endwall slot concepts, the inclination angle should be kept relatively small to introduce the flow as tangentially as possible and energise the boundary layer.

5.3.2 Influence of jet angle

The influence of the jet angle $\beta$ on the performance can be seen in Fig. 10. At the baseline rear-vane suction-side location, maintaining the jet angle $\beta $ within a moderate range of ${0^ \circ }$ – ${20^ \circ }$ yields consistent gains in efficiency and total pressure ratio, with a peak increment in ${\eta _{tt}}$ of $0.385{\rm{\% }}$ at $\beta = {5^ \circ }$ . Larger deflections, such as $\beta = {45^ \circ }$ , as well as negative angles at this location, proved ineffective in suppressing the corner separation and reduced overall performance. The observed variability confirms that ${\eta _{tt}}$ is more sensitive to $\beta $ ( $CoV = 0.220$ ) than ${{\rm{\Pi }}_{tt}}$ ( $CoV = 0.076$ ). Figure 11 displays 3D injector exit velocity streamlines and blade limiting streamlines with surface friction coefficient ${C_f}$ contours for $\beta = {45^ \circ }$ , ${5^ \circ }$ , and $ - {15^ \circ }$ , clearly showing that jets at extreme or negative angles fail to control the separation zone, whereas a moderate positive orientation redirects momentum effectively into the suction-side corner region. While $ = {5^ \circ }$ corresponds to the local peak, these findings emphasise a broader guideline: moderate positive angles directed towards the RV suction side are effective, whereas extremes should be avoided. The key point is to direct smoothly the injected flow towards the separation region in order to energise the endwall flow. Later in the location study it is shown that negative $\beta $ can become beneficial when the injector is positioned on the FV pressure side, highlighting the dependence of the optimal jet angle on slot location relative to the separation zone.

Figure 10. Sensitivity of jet angles to aerodynamic parameters, schematic inset shows definition of $\beta $ .

Figure 11. Blade limiting streamlines with surface friction coefficient ${C_f}$ contours and 3D injector velocity field for different $\beta $ angles.

5.3.3 Influence of radius of curvature

The impact of the injector radius of curvature ( ${R_c}$ ) on injection effectiveness is shown in Fig. 12, where an increase in ${R_c}$ improves performance, consistent with the findings of Dinh and Kim [Reference Dinh and Kim1]. In the present study, the largest tested value of ${R_c} = 12.5$ mm produced the highest increments in ${\eta _{tt}}$ (0.455%) and ${{\rm{\Pi }}_{tt}}$ (0.039%). For a Coandă-type injector, a larger curvature promotes sustained flow attachment on the convex side, enhancing acceleration through the induced radial pressure gradient [Reference Kim, Rajesh, Setoguchi and Matsuo33]. This yields a higher-momentum jet that more effectively mitigates separation zones and improves endwall mixing.

Figure 12. Aerodynamic influence of the injector radius of curvature ${R_c}$ , schematic inset shows definition of ${R_c}$ .

As ${R_c}$ increases, the injector can sustain higher exit velocities, an effect comparable to decreasing the inclination angle $\alpha $ . In this study, values beyond ${R_c} = 12.5$ mm were not investigated, as they produced exit velocities above the specified limits. Moreover, larger ${R_c}$ typically requires higher injection pressures to overcome the increased flow resistance. Overall, the results indicate that using a relatively large radius of curvature is beneficial to promote Coandă attachment and jet acceleration, provided the resulting exit velocity remains within realistic limits.

5.3.4 Role of momentum coefficient

The influence on aerodynamic performance of the circumferential width ${w_c}$ and relative injection mass flow rate ${\dot m_{inj}}/{\dot m_{stall}}$ is analysed in combination due to their strong interrelated effects. From a physical standpoint, an increase in ${\dot m_{inj}}/{\dot m_{stall}}$ with unchanged geometry leads to higher velocity and momentum. Conversely, an increase in ${w_c}$ at a constant ${\dot m_{inj}}$ reduces it. Thus, establishing an optimal balance between ${w_c}$ and ${\dot m_{inj}}/{\dot m_{stall}}$ is crucial.

To this end, we introduce the so-called refined configurations, featuring varying circumferential widths ${w_c}$ to examine the interaction between ${w_c}$ and ${\dot m_{inj}}/{\dot m_{stall}}$ . Accordingly, the inclination angle $\alpha $ is set at ${5^ \circ }$ to strike a balance between aerodynamic advantages and mesh quality. The optimal jet angle $\beta $ of ${5^ \circ }$ was determined based on its standout performance in preliminary analyses. Moreover, the radius of curvature ${R_c}$ is fixed at $12.5$ mm, aligning with the most favourable outcomes. Assumptions for other parameters remain as specified in the baseline scenario. Details of these configurations are presented in Table 5, exploring a spectrum of ${w_c}$ values to evaluate their effects within the predefined ${\dot m_{inj}}/{\dot m_{stall}}$ ranges, selected with consideration for the injector exit Mach number restrictions.

To characterise each injection case, the momentum coefficient [Reference Sarimurat and Dang34], defined as the ratio of the blowing-flow momentum flux to the free-stream momentum flux under the assumption of incompressible flow with equal injection and mainstream densities, is expressed in Equation (4):

(4) \begin{align}{C_u} = \frac{{u_{inj}^2 \cdot {w_c}}}{{0.5 \cdot u_\infty ^2 \cdot {c_{eq}}}} \cdot {\rm{sin}}\left( \beta \right)\end{align}

where ${C_u}$ is the momentum coefficient, ${u_{inj}}$ is the velocity of the blowing flow, ${w_c}$ is the width of the blowing slot, ${u_\infty }$ is the freestream velocity, ${c_{eq}}$ is the equivalent chord length of the tandem stator, measured from the leading edge of the front vane to the trailing edge of the rear vane, and $\beta $ is the blowing (jet) angle. Here, ${u_\infty }$ denotes the mass–flow–averaged freestream axial velocity at midspan of the tandem stator inlet, consistent with the freestream condition used in previous studies [Reference Evans and Hodson32]; in the present setup ${u_\infty } \approx 60{\rm{m\;}}{{\rm{s}}^{ - 1}}$ . This location is annotated in Fig. 3. Since the injector exit Mach number was limited to $Ma \approx 0.35$ (see Section 5.2), density variations are negligible and the simplified form of Equation 4 is valid. Because the jet acts on the near-endwall boundary layer, where the axial velocity is lower than the mid-span freestream, a local normalisation can also be defined. Under equal densities, changing the reference speed to ${u_{local}}$ rescales the coefficient as $C_u^{\left( {local} \right)} = {C_u}{\rm{\;}}{({u_\infty }/{u_{local}})^2}$ . For the smooth-casing case, the axial velocity at the injector location in the upper 10% of the span is ${u_{local}} \approx 35{\rm{m\;}}{{\rm{s}}^{ - 1}}$ , implying $C_u^{\left( {local} \right)} \approx 2.9{\rm{\;}}{C_u}$ ; this rescaling does not affect the observed trends.

Figure 13 illustrates the effects of the refined configurations on aerodynamic parameters for different injection ratios. At a fixed ${\dot m_{inj}}/{\dot m_{stall}}$ , a smaller ${w_c}$ yields better injection effectiveness, whereas at a constant ${w_c}$ , a higher ${\dot m_{inj}}/{\dot m_{stall}}$ enhances performance. Judging by the gains in polytropic efficiency and total pressure ratio, ${\rm{Re}}{{\rm{f}}_4}$ emerges as a strong configuration, achieving the highest increments at the peak ${\dot m_{inj}}/{\dot m_{stall}}$ value (0.49% for ${\eta _{tt}}$ and 0.056% for ${{\rm{\Pi }}_{tt}}$ ). For similar aerodynamic gains, minimising injection rates can be compensated by decreasing circumferential widths.

Table 5. Parameters for refined injection configurations

Figure 13. Interplay of ${w_c}$ and ${\dot m_{inj}}/{\dot m_{stall}}$ on aerodynamic performance.

The equivalent momentum coefficients ${C_u}$ are plotted in Fig. 14 against the ratio ${u_{inj}}/{u_\infty }$ , together with a colormap to show the gains in efficiency and total pressure ratio for each case. The analysis reveals findings similar to those of Sarimurat and Dang [Reference Sarimurat and Dang34], who proposed an analytical model to predict the effectiveness of steady blowing for boundary layer control, assuming incompressible flow of equal density for both main and blowing streams. The study found that a reduction in the momentum thickness of the boundary layer correlates with the momentum coefficient. Relating the reduction of momentum thickness to an improvement in aerodynamic performance, we observed that at a constant momentum coefficient and blowing angle, the gain is more pronounced with higher blown-to-free stream velocity ratios (lower ${w_c}$ ), as long as the blown flow forward velocity exceeds the free stream velocity. In terms of efficiency, for a constant velocity ratio closer to 1, larger ${C_u}$ values represented by larger widths are not beneficial. Conversely, for ${u_{inj}}/{u_\infty }$ closer to 2, the improvement correlates proportionally with the momentum coefficient, as can be seen when comparing the peak cases of every configuration. In terms of the total pressure ratio, higher momentum flow correlates with higher total pressure ratio values for every case of velocity ratio, since we are introducing a higher energy fluid to the main flow.

Figure 14. Relationship of ${C_u}$ and ${u_{inj}}/{u_\infty }$ with aerodynamic performance.

5.4 Enhanced configuration

To ensure comparability across different study locations, it is critical to maintain a uniform geometry. The role of ${w_c}$ and ${\dot m_{inj}}/{\dot m_{stall}}$ was already examined through the refined configurations, where their combined influence was assessed through the momentum coefficient and velocity ratios. Those results indicated that smaller ${w_c}$ values are generally more effective at constant mass flow, but that the benefits can be recovered with higher injection rates. For the subsequent location study, ${w_c}$ is therefore fixed at 10 mm—consistent with the physical space available in the tandem gap—while ${\dot m_{inj}}/{\dot m_{stall}}$ is set at a representative peak value of 0.75%. This choice (Ref ${{\rm{\;}}_3}$ ) enables consistent comparisons between locations, while still reflecting the trends observed in the broader ${w_c}$ – ${\dot m_{inj}}$ parameter space. The resulting configuration shows significant enhancements over the smooth-casing case, with increments of 0.43% in ${\eta _{tt}}$ and 0.047% in ${{\rm{\Pi }}_{tt}}$ .

Figure 15. Comparison of span profiles between smooth, baseline, and enhanced configurations. Left: Stage efficiency, middle: Stage total pressure ratio, right: Static pressure rise for the stator.

Figure 16. Entropy contours at various axial locations and axial velocity contour at $95{\rm{\% }}$ span. Top: Smooth casing case, Bottom: Enhanced configuration.

Span profiles, correlating normalised span $H$ with stage and stator aerodynamic parameters for the smooth casing, baseline, and enhanced configurations, are depicted in Fig. 15. The enhanced configuration outperforms in improving stage efficiency and total pressure ratio, especially near the casing, with the injection effect evident downstream of the injection site above $H = 0.6$ . Accordingly, the static pressure rise coefficient ${C_p}$ for both injection cases is above that of the smooth casing case at near casing regions and, to a lesser extent, at mid and lower span zones. Flow redistribution due to injection slightly influences behaviour beyond the casing endwall region. Specifically, for $H \lt 0.3$ , the near-hub region is minimally affected by the endwall injection. Since there is slightly more pressurised flow in this region for the injection cases, the recirculating flow in the shrouded cavity can slightly increase, marginally raising losses [Reference Hopfinger and Gümmer35]. Nevertheless, the overall impact of the injection remains beneficial.

Furthermore, the entropy contours at different axial locations and the axial velocity contour for $H = 0.95$ are displayed in Fig. 16 for both the smooth casing case and the enhanced configuration. It is evident that the areas of high entropy in the casing corner separation region are considerably reduced, resulting in a smoother flow with diminished vorticity effects. Likewise, the axial velocity contours near the casing reveal a significant reduction in low-momentum zones. The increase in static pressure ratio also induces slightly more recirculating mass flow in the shrouded cavity, leading to increased blockage, as indicated by the entropy distribution near the hub.

5.5 Effects of injector location

This section examines how the location of the injector affects the aerodynamic performance of the compressor. The analysis focuses on the region along the pressure side of the front vane and the suction side of the rear vane. Areas on the suction side of the front vane and the pressure side of the rear vane were found to either not improve or negatively impact performance and are therefore not included in this study. Each location was tested using the same enhanced configuration selected in the previous section and maintained the same relative injection mass flow rate ${\dot m_{inj}}/{\dot m_{stall}}$ . The only variable that changed with the location was the jet angle $\beta $ , with multiple simulations conducted at each location to identify the optimal angle. In this context, ‘enhanced configuration’ refers to a fixed injector geometry ( ${w_c} = 10$ mm, ${R_c} = 12.5$ mm, $\alpha = {5^ \circ }$ ) and injection rate ( ${\dot m_{inj}}/{\dot m_{stall}} = 0.75{\rm{\% }}$ ), while $\beta $ is treated as a location-dependent tuning parameter to ensure the jet is directed towards the local separation zone. Additionally, the same level of mesh refinement for the interface between the injector outlet and stator casing, and the same convergence criteria (as presented in Section 4), were applied across all simulations to ensure consistent results. The final analysed locations are shown in Fig. 17, with the baseline location used in the parametric study highlighted in red.

Figure 17. Final injection locations under study. The baseline location is highlighted in red.

The results of the study of injector locations are detailed in Fig. 18. The efficiency ${\eta _{tt}}$ showed greater variability, with a $CoV$ of 24.68%. The maximum increase in ${\eta _{tt}}$ is observed at an axial location of 1.2 (20% chord of the rear vane), with an efficiency increment of 0.462%. The analysis indicates that performance improvements are attainable in the transition area between the front and rear vanes, especially within the rear vane region. For ${\eta _{tt}}$ increments, the optimal range is found to be between $\zeta $ values of 1 to 1.4; beyond this, efficiency starts to drop. The total pressure ratio ( ${{\rm{\Pi }}_{tt}}$ ) follows a trend very similar to that of ${\eta _{tt}}$ . However, it shows lower sensitivity to changes in injector location with a $CoV$ of 10.86%, as previously found with other studied injection parameters.

The trend of final jet angles, illustrated by the colormap in Fig. 18, shows that the effective $\beta $ range depends strongly on the axial location of the injector. Injections from the FV pressure side require negative $\beta $ to steer the jet towards the RV suction side, thereby improving incidence at the rear-vane leading edge and mitigating the endwall separation. In contrast, injections from RV suction-side positions demand positive $\beta $ , since these sites are already close to the loss region and benefit from directing momentum into the corner deficit. The magnitude of the required jet angle scales with proximity: locations further upstream need larger angles, whereas slots closer to the separation onset require only small adjustments. For example, angles of about ${20^ \circ }$ are effective at $\zeta = 1.2$ , while values near ${5^ \circ }$ suffice further downstream. These results confirm the jet-angle sensitivity guideline, showing that performance is maintained across a moderate range of $\beta $ values as long as the jet targets the separation zone, with $\beta $ naturally adapting to injector placement rather than being fixed to a single value.

Figure 18. Efficiency and total pressure ratio variations with injector location. The colormap indicates the jet angle ( $\beta $ ), highlighting regions of maximum aerodynamic benefit between the front and rear vanes.

Building on this trend, Fig. 19 compares the effect of the selected jet angles across two representative injection positions at $\zeta = 0.5$ and $\zeta = 1.2$ . For $\zeta = 0.5$ , the best performance is obtained by directing the flow towards the leading edge of the rear vane; however, because the injection is located far upstream of the separation onset, the jet diffuses and loses most of its momentum before reaching the critical region. In contrast, at $\zeta = 1.2$ , which lies just upstream of the separation point, the boundary layer is more effectively removed. Such conclusions, previously reported for configurations with single aerofoils [Reference Fottner36], are here extended to tandem vanes, providing the first evidence of these effects in published literature.

Figure 19. Surface wall and 3D injector velocity streamlines, alongside the axial velocity contour at the stator outlet for two cases of $\zeta $ . The jet angle $\beta $ changes accordingly to the axial location of the injector.

For the subsequent analysis, distinctive cases identified in the study of injection locations are selected. Specifically, two front vane locations at $\zeta = 0.1$ and $\zeta = 0.5$ are chosen to illustrate lower and mid-range performance, respectively. In contrast, rear vane locations at $\zeta = 1.05$ (near the leading edge) and $\zeta = 1.2$ are selected for their notable impacts on ${\eta _{{\rm{tt}}}}$ and ${{\rm{\Pi }}_{{\rm{tt}}}}$ . Further locations at $\zeta = 1.4$ and $\zeta = 1.6$ are included to characterise behaviour at and beyond the separation point. These selections are detailed in Table 6 and will be used for a detailed analysis of span profiles.

Table 6. Selected injection location cases for detailed analysis

Figures 20 and 21 showcase the span profiles for ${\eta _{tt}}$ and ${{\rm{\Pi }}_{tt}}$ for selected injection location cases, highlighting divergent behaviours above 85% span. The impact of injection becomes evident beyond 60% span, with all cases showing increased values compared to the smooth casing scenario. For efficiency, cases near the rear vane leading edge ( $\zeta = 1.05$ and $\zeta = 1.2$ ) exhibit a uniform and significant increase across a wider span, suggesting enhanced mixing. Specifically, $\zeta = 1.05$ exhibits a consistently smoother efficiency rise, whereas $\zeta = 1.2$ shows superior performance at lower span values, diminishing towards $H = 1.0$ . For the total pressure ratio, the rear-injection cases show localised differences confined to the near-casing region, with ${{\rm{\Pi }}_{tt}}$ peaking close to $H = 1.0$ . In particular, $\zeta = 1.2$ exhibits a distinct peak around $H = 0.9$ , indicating concentrated injected flow at this location. This profile behaviour aligns with the jet angle $\beta $ for this case, set at ${20^ \circ }$ , since larger $\beta $ values direct the jet more strongly toward the blade surface and shift the high-energy flow towards lower span zones. These variations are highlighted by the zoomed x-axis in Fig. 21 but become less pronounced once mass-flow averaging is applied, as seen later in the speedline results.

Figure 20. Spanwise distributions of polytropic efficiency for highlighted injection cases. A zoomed-in view shows behaviour between $H = 0.85$ and $1.0$ .

Figure 21. Spanwise distributions of total pressure ratio for highlighted injection cases. A zoomed-in view shows behaviour between $H = 0.85$ and $1.0$ .

Overall, for larger $\zeta $ values beyond the separation point, the influence of injection is confined to a smaller span region due to diminished mixing. Near the hub, at $H = 0.4$ , flow redistribution minimally affects dynamics around the shrouded cavity for all cases.

5.6 Effect of injection on tandem stator loading

In this section, the impact of air injection on blade loading is analysed at the design point condition. Figure 22 compares the smooth casing and a representative injection case at $\zeta = 1.2$ , both evaluated at $H = 0.9$ . The blade loading profiles of $M{a_{is}}$ are plotted against the normalised chordwise location $s$ , and additional aerodynamic parameters such as turning ${\rm{\Delta }}\theta $ and the flow angle at the stator outlet ${\theta _{out}}$ are presented. As observed in the $M{a_{is}}$ curves, the induced loading due to injection is evident in the rear vane compared to the smooth casing case, showing the removal of corner separation near the casing. This removal of low-momentum flow unlocks additional diffusion capacity in the rear vane.

Figure 22. Injection impact on aerodynamic loading. Left: Blade loading profiles at $H = 0.9$ comparing the smooth casing and an injection case with $\zeta = 1.2$ , showing the isentropic Mach number. Right: Turning ${\rm{\Delta }}\theta $ and outlet flow angle ${\theta _{out}}$ profiles at the stator outlet for both cases.

The overall mass-flow averaged diffusion factor $DF$ for the injection case is 0.54, compared with 0.521 for the smooth casing. The increase is entirely attributable to the rear vane, with an independent value of 0.29 for injection versus 0.27 for the smooth casing. Consequently, the turning ${\rm{\Delta }}\theta $ of the stator row increases, and the outlet flow angle ${\theta _{out}}$ shows higher values for $H \gt 0.6$ , indicative of the enhanced rear vane loading. Correcting underturning with air injection not only improves blade loading and stage efficiency but also benefits the incidence for subsequent stages and, therefore, overall efficiency in multi-stage compressors.

5.7 Impact at off-design conditions

We present the impact of the selected configurations under off-design conditions. The relative injection mass flow rate, ${\dot m_{inj}}/{\dot m_{stall}}$ , is kept constant, which leads to variations in the total pressure of the injected fluid across different operating points, from de-throttled to near-surge conditions. This variation correlates with an increase in static pressure at the injection site, typically seen when the injector is fed by downstream flow. The simulations start at the design point and systematically progress towards de-throttled and surge conditions by adjusting the outlet mass flow rate. As a reminder, the speedlines presented in this section were obtained using the ‘equal inlet mass flow’ method described in Section 4, imposing the outlet mass flow rate but setting its value so that the resulting inlet mass flow matches the smooth-casing case. In this way, direct comparability between the smooth-casing and injection cases is ensured at each operating point.

For all injection cases, stage efficiency ${\eta _{tt}}$ exceeds that of the smooth-casing scenario in Fig. 23. Notably, the cases near the rear vane consistently outperform those at the front vane across the speedline. The incremental benefit follows a clear trend: it decreases towards throttled/near-surge conditions and increases towards de-throttled conditions. For example, the increment of the enhanced case ( $\zeta = 1.4$ ) at $\dot m/{\dot m_{DP}} \approx 1.10$ is about $ + 0.78{\rm{\% }}$ , larger than the $ + 0.43{\rm{\% }}$ observed at the design point. This behaviour is consistent with the mechanism: at higher mass flow, negative incidence on the front vane strengthens the endwall deficit convected into the tandem gap, enlarging the suction-side separation of the rear vane in the smooth-casing baseline. Injection mitigates this effect, recovering a greater fraction of the loss. Beyond the de-throttled regime, the profiles gradually converge and approach the smooth-casing curve.

Figure 23. Comparative impact of injection location on stage polytropic efficiency (left) and total pressure ratio (right) under various operating conditions.

Conversely, the total pressure ratio differences between locations are minimal, with all injection cases outperforming the smooth-casing scenario. It is also important to note that while injection in the tandem stator does not extend the stall range-predominantly determined by rotor tip flow—it enhances stage performance across the entire operating range.

5.8 Sensitivity to turbulence model

The baseline simulations in this study were performed with the SST turbulence model, consistent with prior FRANCC numerical setups. While SST is standard in compressor CFD, it is known to have limitations in predicting streamline curvature effects and reattachment in separated flows. Nevertheless, SST has been shown to reproduce tandem-stator separation patterns in related low-speed compressor experiments [Reference Tesch, Lange, Vogeler, Ortmanns, Johann and Gümmer11], so it provides a reasonable baseline. To assess the robustness of our findings, two alternative models were also tested: SST with the reattachment modification (SST+RM) and SST with the $\gamma $ –Re ${{\rm{\;}}_\theta }$ transition model [26]. These comparisons were restricted to a single representative case—the enhanced configuration—at the design point and, where feasible, along the speedline.

For consistency, the same turbulence intensity of 5% was applied at the injector inlet as in the baseline setup. Since the turbulence level of the jet could in principle influence reattachment by enhancing local mixing, injector turbulence intensity was additionally varied between 1%, 5% and 10% for the enhanced configuration. The resulting increments in total pressure ratio were identical within solver output precision, while the changes in efficiency were negligible: the coefficient of variation across the cases was below ${10^{ - 4}}$ for all models, much smaller than the reported performance increments (0.1–0.6%). This confirms that the dominant factor for reattachment is the injected momentum flux rather than the prescribed turbulence intensity.

A quantitative design-point summary is given in Table 7, reporting the increments of the enhanced configuration relative to the SC for the three models. As seen in Table 7, the alternative models predict similar increments in total pressure ratio, while the efficiency gain is reduced compared with the baseline SST. Since the $\gamma $ –Re ${{\rm{\;}}_\theta }$ model produced design-point results nearly identical to the reattachment modification but encountered convergence difficulties at off-design operating points, its results are included in the table for comparison only. Although the mesh resolution was sufficient, these convergence issues are not unexpected, as the transition model introduces additional transport equations and can be sensitive in strongly three-dimensional separated flows. At the design point, the $\gamma $ –Re ${{\rm{\;}}_\theta }$ result indicated $\gamma \approx 1$ throughout the passages, confirming fully turbulent boundary layers. Local values of $\gamma \approx 0.02$ were present in the wall-adjacent cells as part of the near-wall treatment described in the ANSYS CFX-Solver Modelling Guide⁽²⁶⁾. The detailed flow-field comparison below focuses on SST vs. SST with reattachment modification.

Table 7. Design-point increments of the enhanced configuration relative to the SC

Figure 24 compares the baseline SST (top row) and SST with the reattachment modification (bottom row). For each model, the left block shows the SC and the right block the enhanced configuration. Within each block, the left subpanel plots axial-velocity contours on the spanwise plane $H \approx 0.95$ , whereas the right subpanel shows the suction-side skin-friction coefficient ${C_f}$ with limiting streamlines over the full surface; the dashed line on the ${C_f}$ plots marks where the $H \approx 0.95$ plane intersects the blade. For the smooth casing, SST predicts a larger suction-side separation bubble, a broader low ${C_f}$ region, and stronger streamline divergence; SST+RM yields a shorter separated region with earlier reattachment. With the enhanced injector, both models produce very similar patterns: the suction side exhibits higher ${C_f}$ , the limiting streamlines remain attached, and the corner region is stabilised. This confirms that the injection mechanism remains effective regardless of the turbulence model.

Figure 24. Rear vane flow for SST (top) and SST+RM (bottom). Each block shows axial velocity contours on $H \approx 0.95$ span plane and suction-side ${C_f}$ with limiting streamlines for the smooth casing and enhanced configurations. Flow direction is indicated by arrows.

The differences in predicted increments are consistent with each model’s eddy-viscosity behaviour under adverse pressure gradients. Plain SST generates relatively high turbulent viscosity in the separated shear layer, which diffuses momentum gradients and tends to exaggerate the extent of separation. The reattachment modification reduces this overproduction, yielding thinner shear layers and stronger natural entrainment of high-momentum fluid into the bubble, and thereby smaller pockets. While this explains the lower absolute gains in $\eta $ , the central conclusion remains unchanged: in all models, injection acts as an artificial entrainment mechanism that robustly suppresses the suction-side separation. In the absence of experimental reference for this preliminary geometry, these results are best interpreted as a robustness assessment of the modeling approach.

To extend the comparison beyond the design point, Fig. 25 shows speedlines for the reattachment-modified SST model. The trends are consistent with those seen for the baseline SST: the enhanced configuration outperforms the smooth casing across the entire operating range, although the absolute increments vary. At the design point the efficiency gain is about $ + 0.13{\rm{\% }}$ , but toward de-throttled conditions the improvement increases, reaching $ + 0.36{\rm{\% }}$ at $\dot m/{\dot m_{DP}} = 1.1$ and $ + 0.61{\rm{\% }}$ at $1.15$ . This reflects the flow physics at these conditions, where negative incidences amplify suction-side separation of the rear vane as already discussed in Section 5.7. In contrast, increments diminish toward throttled conditions, where smooth-casing and injection cases converge. For the total pressure ratio, the behaviour follows the same trend but with smaller relative differences. Overall, while the absolute magnitudes of ${\eta _{tt}}$ and ${{\rm{\Pi }}_{tt}}$ vary with the turbulence model, the reduction of suction-side separation and the associated performance gains are consistently predicted, confirming that injection provides a robust means of mitigating endwall-driven corner separation, with the strongest benefit under conditions where separation is most severe.

Figure 25. Speedlines for SST with reattachment modification: comparison of SC and enhanced configuration.

5.9 Injection losses

Throughout this study, our goal was to analyse the aerodynamic impact of injection in a tandem stator configuration and understand how an injection strategy characterised by the momentum coefficient ${C_u}$ correlates with improvements in stage efficiency and total pressure ratio. Therefore, the source of the injected mass and the potential penalties incurred were not subjects of study. However, in realistic operational conditions, the injection implementation can provoke additional losses in the system.

One common approach to account for total losses, including those from injection, is presented in the work of Evans et al. [Reference Evans and Hodson32]. According to this study, the total pressure loss coefficient, denoted as ${\omega _t}$ , is defined by a mass-weighted approach that accounts for both the main flow and injected jet contributions. The total pressure loss coefficient is given by:

(5) \begin{align}{\omega _t} = \frac{{{{\dot m}_{in}}\left( {\overline {{P_{t,in}}} - \overline {{P_{t,out}}} } \right) + {{\dot m}_{inj}}\left( {\overline {{P_{t,inj}}} - \overline {{P_{t,out}}} } \right)}}{{{{\dot m}_{in}}\left( {\overline {{P_{t,in}}} - \overline {{P_{in}}} } \right) + {{\dot m}_{inj}}\left( {\overline {{P_{t,inj}}} - \overline {{P_{inj}}} } \right)}}\end{align}

where ${\dot m_{in}}$ and ${\dot m_{inj}}$ denote the mass flow rates of the main and injected streams; $\overline {{P_{t,in}}} $ and $\overline {{P_{t,out}}} $ are the mass-flow-averaged total pressures at the tandem stator inlet and outlet, as shown in Fig. 3; and $\overline {{P_{t,inj}}} $ is the mass-flow-averaged total pressure at the injector inlet. This relationship takes into account the total pressure losses with respect to the total kinetic energy in the tandem stator domain. When ${\dot m_{inj}}$ is equal to zero, we obtain the standard definition of the total pressure loss coefficient $\omega $ .

Taking Equation (5) into consideration, we present an example illustrating an injection strategy that accounts for injection losses. As before, the normalised axial location is set at $\zeta = 1.4$ . However, this time a circumferential width ${w_c}$ of 6 mm and an injection rate ratio ${\dot m_{inj}}/{\dot m_{stall}}$ of 0.25% are chosen to minimise the generated losses. This configuration results in a momentum coefficient ${C_u}$ of 0.01 and a ratio ${u_{inj}}/{u_\infty }$ of 1.2, in contrast to the enhanced configuration presented in Section 5.4, which had ${C_u} = 0.05$ and ${u_{inj}}/{u_\infty } = 2.1$ .

As shown previously in Fig. 16, the reduction of entropy along the axial direction after the injection location indicates effective mixing of the injected flow with the main flow. However, the entropy contours at the stator casing, shown in Fig. 26, reveal an increase in entropy due to the injection itself. This requires consideration of additional losses using Equation (5). For the larger momentum coefficient case ( ${C_u} = 0.05$ ), the total pressure loss coefficient ${\omega _t}$ is 0.0494, which is slightly higher than in the smooth casing case. In contrast, for the smaller momentum coefficient case ( ${C_u} = 0.01$ ), the calculated total pressure loss coefficient ${\omega _t}$ is 0.04396, compared to 0.04350 when injection effects are not considered, indicating that the injection-induced losses are negligible. Despite this, the calculated ${\omega _t}$ still represents a reduction of 5.1% compared to the smooth casing case, where $\omega $ is approximately 0.0462. Additionally, an increase in polytropic efficiency ${\eta _{tt}}$ of about 0.28% is observed.

Figure 26. Comparison of the effects on static entropy at the stator casing for the smooth casing case and different injection strategies: small ( ${C_u} = 0.01$ ) and larger ( ${C_u} = 0.05$ ) momentum coefficients.

Figure 27 shows the comparison of this injection configuration ( ${C_u} = 0.01$ ) with the smooth casing case and the equivalent higher momentum coefficient configuration ( ${C_u} = 0.05$ ). A lower momentum coefficient configuration yields a lower increment but still offers aerodynamic improvements. As seen in the efficiency and total pressure ratio profiles, the main difference is the behaviour at span values $H \gt 0.9$ , very close to the casing, where the higher momentum jet shows a pronounced influence. However, the effect of unlocking more diffusion in the rear vane of the tandem stator remains with smaller injection rates, as observed in the blade loading profiles. This approach presents a conservative method to account for injection losses. However, further adjustments may be necessary for each specific application. For instance, utilising a recirculation channel to extract air from the endwall region of the second-stage rotor and re-inject it into the tandem stator in the first stage could enhance performance in other critical areas of the compressor, such as mitigating rotor tip leakage. This highlights the importance of an integrated approach to flow control evaluation, rather than relying on isolated assessments.

Figure 27. Comparison of injection strategies with small ( ${C_u} = 0.01$ ) versus larger ( ${C_u} = 0.05$ ) momentum coefficients to minimise injection losses, for cases at $\zeta = 1.4$ . Left: Blade loading profiles at $H = 0.9$ showing the isentropic Mach number. Right: Polytropic efficiency and total pressure ratio profiles.