Experimental and Computational Investigation of an Advanced Casing Treatment in a Single-Stage High-Speed Compressor

Axial compressors used in propulsion applications may be at risk of a fluid-dynamic instability known as stall. This generally occurs at off-design conditions when the mass flowrate is too low for stable operation. In such cases, the compressor pressure rise, mass flowrate, and efficiency may decrease dramatically. In some instances, stages of a compressor may stall transiently during normal operation. In other cases, violent surge may occur, requiring engine shut down and repair. Compressors are therefore designed with a stall margin (SM) such that the design mass flowrate is sufficiently high compared with the stall point. Attempts to understand the compressor stalling phenomenon have been accompanied by efforts to increase the stable operating range of compressors (e.g., Ref. [1]).

This paper provides results and analysis from a detailed experimental and numerical study of an advanced casing treatment (CT). The term CT refers to modifications to the casing around the compressor rotor. The intent is to passively alter the fluid mechanics of the tip region of a compressor to improve the stall margin. However, CTs add manufacturing cost and generally are understood to reduce the efficiency of the compressor. Moreover, design is often a trial-and-error approach, with the adoption of CTs often limited to cases where an operational problem has been identified.

The current era of advanced CT research can be traced back to the parametric experimental studies of the 1970s. A wide range of porous casing or passive geometries were studied in rig tests including perforated plates, honeycombs, circumferential grooves, axial slots, and tangentially skewed axial slots. A common characteristic of these early experimental studies is that the entire tip chord of the rotor was covered by the porous wall treatment, in some cases covering more than the axial chord projection of the rotor tip. Multiple geometries of casing treatment designs were studied, such as the skewed axial slot or the circumferential groove, to measure the impact of CT on compressor performance. Overall, many numerical and experimental studies have shown that recirculating casing treatments can significantly improve the rotor stall margin [2–4]. A comprehensive review of CT technology is provided by Hathaway [5].

The early studies of CTs resulted in many learnings, including the discovered efficiency penalty for a given stall margin improvement. Experimental data from Fujita and Takata [6] supported this view. Cumpsty estimates these early CTs resulted in $≈1%$ efficiency penalty for a 10% stall margin improvement [7]. Treatments effective for high inlet relative Mach number also worked for low speeds. So, desirable features can be obtained from studies at both high and low speeds. Circumferential grooves had a lesser impact on stall margin and efficiency, but were the simplest CT design. Axial-skewed slots gave the most stall margin improvement but with significant efficiency penalty (⁠$>1%$⁠). Tangential leaning axial slots in the direction of rotation provided the best stall margin improvement. In the early 2000s, Seitz [8] moved the slot forward (upstream of the blade). This reduced the recirculation flow through the CT and efficiency penalty. Interest increased with experimental results showing variations on axial-skewed slots with significant stall margin increases and little efficiency loss [8,9].

The current versions of advanced CTs build on many of the results mentioned above, essentially expanding on the skewed axial slot geometry to include additional geometrical parameters. The use of unsteady computational fluid dynamics (CFD) analysis during the design process has allowed exploration of the effects of number of slots, solidity, slot position, length, and arbitrary control of slot shape. The present effort was focused on a CT design that can be characterized as a bent-skewed slot. A diagram of this CT design is shown in Fig. 1. The test article was a high-speed, single-stage, axial compressor that is typical of the front stage of a modern core compressor. Computations and experiments indicated the potential for a significant increase in stall margin and pressure ratio, with little-to-no change in the design point efficiency.

This paper explores three main objectives related to compressor performance with a smooth wall (SW) configuration and a CT configuration. The first objective of this study was to determine the efficacy of the designed CT. This would be accomplished by completing an experimental test campaign focused on acquiring compressor performance data for a compressor with and without CT. The second objective was to verify the use of unsteady Reynolds-averaged Navier–Stokes (URANS) modeling for predicting CT effects on compressor performance data. Detailed comparisons between URANS results and experimental measurements will be shown. Finally, a combination of detailed experimental measurements and validated computational results would be used to improve understanding of the fluid mechanics that are influenced by the presence of the CT. Special attention was given to the flow entering and exiting the CT, and the interaction this flow has with the surrounding tip-leakage flow.

The experimental results presented in this paper were obtained with the Notre Dame front stage core compressor (ND-FSCC) facility. The facility was designed for continuous operation of front stage axial core compressors and has been utilized in a wide variety of compressor aerodynamics studies (e.g., Refs. [10,11]). The facility flow path supported mass flows up to 23 kg/s (50 lbm/s). The compressor operating point was set by valves upstream and downstream of the test article. The facility inlet system drew air from the atmosphere, through a filter system and mixing chamber, then passed through a calibrated Venturi which provided a measurement of the physical mass flowrate. The flow moved past the upstream valve and through inlet screens that conditioned the flow to be circumferentially and radially uniform. The flow then entered the inlet duct and test article, then continued out through an exit duct. The facility discharge system had the capability to recycle part or all of the exhaust back to the inlet system to raise the inlet temperature. Otherwise, the flow was discharged to the atmosphere. The compressor rotor shaft was supported by active magnetic bearings and driven by a 500 kW (670 hp) electric motor through a speed increasing gearbox. The motor and gearbox were capable of producing maximum shaft speeds of roughly 27,500 rpm. The hub-to-tip ratio of this rotor was roughly 0.5.

Figure 2 shows the cross section of the stage. Flow moves from the left of the figure to the right and exits through the outlet guide vanes to the discharge system described previously. The test article consisted of a transition duct, inlet guide vane (IGV), rotor (R1), and stator (S1). The transition duct section nominally represents the transition from the exit of the low-pressure compressor and transitioning to the smaller diameter of the high-pressure compressor. The IGV and S1 airfoils were assembled with an automated actuation system in order to control their respective stagger angles. Additionally, a locking mechanism was used to fix the stagger angle to avoid uncertainty and repeatability issues related to the actuation.

Key measurement planes are labeled in Fig. 2. Plane “1” is located at the inlet to the test article. Plane “2” is approximately halfway through the transition duct. The rotor-exit plane and stage exit plane are denoted as “3” and “4”, respectively.

The test facility utilized standard data acquisition systems and high-accuracy instrumentation for low sampling frequency data, such as temperature and slow response pressure measurements. Total pressure/total temperature “rakes” were distributed throughout the test article. These rakes were installed in the flow path at multiple circumferential positions at each key measurement plane. These rakes account for the total pressure and temperature measurements acquired in the test article. There were also several static pressure measurements at the casing wall at each key measurement plane. Instrumentation on the rotor case included four tip clearance sensors that measured rotor blade tip to case distance. Finally, light probe sensors were also utilized on the case to capture blade deflection data and monitor aeromechanical health of the rotor blades during a test.

The compressor shaft rotational rate was acquired via an optical 1/rev sensor. A relative humidity sensor was positioned upstream of the test article, which allowed for humidity corrections in the data post-processing. Furthermore, high spatial-resolution traverse systems were used for spanwise and/or circumferential surveys at select measurement planes, including the rotor and stage exit. Finally, shaft torque measurements were acquired via a temperature compensating, telemetry-based torque meter.

Rotor pressure ratio was computed from area averages from the respective planes. Temperature ratio was computed using both area average values from the thermocouple rakes and from the torque measurement. All results were then normalized by the design point values. Standard uncertainty analysis computations were completed. In addition, several build-build repeat tests were conducted to understand the repeatability of the measurements. Relevant uncertainty and repeatability values are presented in Table 1.

Post-test CFD analysis of the experiments was conducted with two objectives. The first objective was to validate the ability of URANS simulations to obtain accurate predictions of the smooth wall and casing treatment performance parameters. The second objective was to use these validated computational results to improve understanding of the change in flow physics due to the casing treatment.

Figure 3 shows the part-wheel domain and setup used in the post-test CFD analysis. All three blade rows (IGV, rotor, and stator) were included in the CFD domain. IGV:R1:S1 blade counts of 4:3:6 respectively were modeled in the part-wheel CFD domain. Periodic boundary conditions were applied across the part-wheel sector.

The URANS simulations were performed using commercial flow solver ansys-cfx. The RANS model chosen was the Wilcox k–ω turbulence model with Launder–Kato production limiter. In terms of spatial discretization, a second-order accurate discretization of flow equations and first-order discretization of turbulence equations were used. CFD grid resolutions were designed to accurately capture relevant flow features, especially for loss prediction. The total grid size for all blade rows including CT was 20.3 million. Air was modeled as a perfect gas with constant γ = 1.40037 and constant c_p = 1.006 kJ/kg K $(6026.76ft2/s2R)$⁠. Sutherland’s law was used for viscosity. All interfaces between blade row were sliding mesh. All simulations were run until measured parameters remained essentially constant with increased iteration count.

Post-test CFD was setup to closely match the experimental operating conditions using available measurements. At each operating point, measured conditions such as Rotor1 tip-gap as well as IGV and Stator1 setting angles were closely matched between experiments and CFD. Rotor1 had a non-axisymmetric hub that was also modeled in the CFD. Stator1 seal leakage flows were modeled as source terms within stub-cavities upstream and downstream of Stator1. Source term levels applied in the unsteady RANS were determined using a simplified seal model included in an accompanying smooth wall steady RANS calculation. The location and width of the seal leak cavity (front) were altered to suit the sliding mesh interface between rotor and stator.

Data-matched through-flow analyses were used to specify boundary conditions at IGV inlet and Stator1 exit. Measured radial profiles were not available at IGV inlet. So, at chosen operating points, through-flow analyses were calibrated to match available test data. Rig performance data and radial profiles were matched at stations where test data were available. This data-matched analysis was used to specify total pressure, total temperature, and flow angle profiles at the IGV inlet. Inlet turbulence profiles to the URANS were extracted from a steady mixing-plane RANS simulation at the smooth wall design point. The steady RANS domain extended from the upstream inlet to Stator1 exit. Lastly, at the exit of the URANS domain, a static pressure boundary condition was imposed with a profile shape based on radial equilibrium. Exit static pressure level was manually altered to match the desired Rotor1 operating line.

Experimental data were acquired for two rig configurations: an SW rotor case and a CT rotor case. Characteristics were acquired for each configuration at four corrected speed settings: 88%, 92.5%, 95%, and 98%. Experimental pressure ratio and temperature ratio for the four speed lines of each configuration are shown in Figs. 4 and 5, respectively. In these figures, CT data are plotted with squares while SW data are plotted with circles. The corrected speed corresponding to each set of data is denoted by the percent label on the figure. All values were normalized by the 98% speed nominal design point.

An increase in corrected speed is observed to increase rotor pressure ratio and temperature ratio, as expected. For each characteristic, pressure ratio and temperature ratio increase as corrected mass flow decreases until the stall point was reached. At each corrected speed, the CT was found to increase the corrected mass flow, the pressure ratio, and temperature ratio of the compressor. Finally, the stalling corrected mass flow decreases for each speed when comparing the two configurations. The CT allows for the stage to be operated at lower corrected mass flows, and therefore have a larger range of stable operation than the SW case.

It is of interest to examine the 98% results in detail. Figure 6 shows π_r as a function of $m˙c$ for the two characteristics at 98% corrected speed. For future use, labels of four important characteristic locations are added to this figure to give the reader clarity when describing speed line condition: “CH” denotes choked flow location, “OL” shows operating line condition, “SWNS” identifies the smooth wall near-stall condition, and “CTNS” is the casing treatment near-stall condition.

At the OL condition, the CT data show a 0.76% increase in π_r and roughly a 0.2% increase in $m˙c$ when compared to SW values. At the SWNS condition, the increase in π_r is 1.5% and the increase in $m˙c$ is around 1.3% from SW to CT. SM can also be computed to determine the increase in stable operating $m˙c$ range by the introduction of the CT. For the SW configuration, SM was 9.9%. With the CT case, SM was 19.3%.

The stage efficiency was calculated from the pressure ratio and temperature ratio values discussed. Figure 7 shows the stage efficiency computed for SW and CT configurations at the four measured corrected speeds. The 98% characteristic shows that the efficiency of the CT configuration was slightly higher than the SW values. Quantitatively, at the design point, CT efficiency was determined to be 0.5% higher than SW efficiency. This difference increases to 1.2% at the SWNS point. The 95% speed results show a larger increase in efficiency with the CT compared with the SW. Specifically, an increase of 1.1% was observed at the near-stall point, with a maximum change of 1.7% observed near the local maximum. The efficiency at both 88% and 92.5% speeds were also affected by the CT, although the data indicated lower efficiencies at lower mass flowrate for each speed.

The experimental measurements were used to compare with the results of the numerical simulations in order to validate the computations. The CFD was sampled at the same spatial locations as the measurement probes, and averaged to allow comparison with the experiment. Pressure ratio and corrected mass flowrate values are shown for CFD and experiments in Fig. 6. The diamond markers denote SW and CT URANS results.

The SW computations show a slight overprediction in π_r and $m˙c$ at both OL and SWNS conditions. π_r was overestimated by approximately 0.9%, while the $m˙c$ overestimate was approximately 0.6%. With the addition of the casing treatment, the OL CFD calculation predicted only a slight increase in $m˙c$⁠, with almost no change in π_r. This resulted in a predicted mean operating point that was in near-perfect agreement with the CT experimental characteristic. Similarly, the CFD at the CTNS condition agreed with the experiment to within the experimental uncertainty.

The spanwise distribution of pressure and temperature ratio were also considered, as these provide insight into the CT flow physics as well as a more detailed validation of the numerical simulations. These results are shown in Figs. 8–11. In these figures, rotor-exit total pressure or total temperature ratio is shown for a given condition of the SW and CT configurations. Percent span is plotted on the ordinate, while total pressure ratio or total temperature ratio is plotted on the abscissa. The rotor hub is defined as 0% span while 100% span is the rotor case. The black dots indicate experimental measurements, while the solid black line denotes the CFD-predicted values.

In Fig. 8, rotor-exit total pressure ratio is computed and plotted for the OL condition of the SW and CT configurations. The SW OL data are shown in the top plot, while the CT OL data are shown in the bottom plot. In this figure, grid spacing for pressure ratio is 0.2. For the SW data, the CFD results follow the experimental values well. The CT data generally agree well, although the experimental values measure a higher total pressure ratio at the tip than the CFD values.

The corresponding total temperature ratio is observed in Fig. 9 for the SW and CT OL condition. Grid spacing for the total temperature ratio for this figure is 0.05. Similar to the total pressure ratio data, the CFD and experiments agree well for the SW configuration, while the CT configuration sees lower CFD-predicted values towards the rotor tip compared to experiments.

The experiment and CFD-predicted spanwise values were quantitatively compared to each other for the OL condition. The computed CFD radial profiles were interpolated to the experimental spanwise measurement locations. The difference in experiment to CFD value at each spanwise location was then calculated as $QCFD−QExpt$⁠, where Q is the quantity of interest (pressure ratio or temperature ratio). The mean and standard deviation of the calculated differences were then computed to quantify the mean bias and noise of the CFD predictions compared to experiments. Total pressure ratio data for the SW OL configuration compute a mean difference of 0.0078 and a standard deviation of 0.0147. The mean and standard deviation of the SW OL total temperature ratio comparison data were −0.0025 and 0.0037, respectively. The CT OL total pressure ratio mean difference was computed as −0.0105, while the standard deviation was found to be 0.0267. The CT OL total temperature ratio mean difference was -0.0078 and the corresponding standard deviation was 0.0079. These presented total pressure and temperature ratio comparison values quantify the good agreement observed between URANS using cfx and experiments at the OL condition for both SW and CT configurations.

Spanwise rotor pressure ratio is shown in Fig. 10, this time for the NS condition of each of the SW and CT configurations. Total pressure ratio grid spacing is 0.1. In this figure, the experimental values are seen to have a greater spread as compared to the OL condition. The CFD and experiment values agree well, with the CT configuration CFD slightly under-predicting total pressure ratio toward the rotor tip.

For completeness, spanwise total temperature ratio for the SW and CT NS condition is given in Fig. 11. The grid spacing of total temperature ratio is 0.05. The computed total temperature ratio of the CFD and the experiments agree very well for both the SW and CT configuration at the NS condition.

Rotor-exit traverses of total pressure and total temperature were obtained in order to provide additional detail regarding the effects of operating condition and casing configuration. The probe was traversed across the span at the four operating conditions: CH, OL, SWNS, and CTNS for both SW and CT (CTNS was only acquired with the casing treatment, as the smooth wall rig could not be safely run in this condition). The rotor corrected speed was set to 98%. The pressure ratio and temperature ratio are shown in Figs. 12 and 13, respectively. The rotor-exit span s is represented by the ordinate with the corresponding π(s) or τ(s) on the abscissa. The solid lines correspond to data acquired with the smooth wall configuration, while the square symbol lines correspond to data acquired with the casing treatment configuration. All four conditions for each configuration are plotted on the figures: CH, OL, SWNS, and CTNS. The associated condition is also labeled on the figures.

Figure 12 shows the measured rotor-exit total pressure ratio data. The pressure ratio at CH conditions was found to be approximately π = 1.05 near the hub, and decreased in value across the entire span towards the tip, where π(s) ≈ 0.8. Differences between the SW and CT configurations were minimal for s < 0.7. A small increase in CT π(s) was observed for s > 0.7. This suggests the CT was not having a significant impact on the flow overall at this operating condition. This could be anticipated, as the pressure differences across the blades are small at the CH condition where the pressure ratio is small (especially near the tip as noted in the spanwise profile).

The pressure ratio values for the OL condition were found to be similar to the CH condition at the hub, but decreased only slightly across the span to π(s) ≈ 1. Changes to the pressure ratio were observed to be minimal when comparing SW to CT. While large changes in the pressure ratio between the SW and CT conditions were not expected, it is perhaps surprising that the differences are even smaller than the CH condition. That is, the blade (and tip region in particular) was operating at near design blade loading, which might have been expected to cause significant interactions with the casing slots. The data, however, show little changes, which suggests that the CT slots were not altering the flow significantly.

The SWNS data (in green) show a pressure ratio near the hub that is slightly increased from the CH and OL results. The SW data show a decrease in π from 0 < s < 0.6, with a local minimum followed by an increase in pressure ratio from 0.6 < s < 1.0. The CT results show a nearly identical pressure ratio (compared with SW) near the hub, but also a nearly constant value across the entire span. Interestingly, the CT increases the observed pressure ratio from 0.4 < s < 0.92, and then shows a decrease in value near the casing.

The CTNS condition shows a pressure ratio that is again slightly higher at the hub compared with the higher mass flow conditions. The values across the span are relatively constant, and then increase slightly towards the outer part of the span.

The rotor-exit traverse total temperature ratio data are shown in Fig. 13. Note that the interpretation of temperature ratio is often linked directly to the flow turning of the compressor. The CH condition data show a temperature ratio that is relatively constant from 0 < s < 0.4 with a small increase in values for the CT case. At higher spanwise locations, τ(s) can be observed to decrease to a local minimum, followed by an increase near the tip for both SW and CT conditions, although the CT values were generally slightly higher than those from the SW. These results, in combination with π(s), suggest that the CT does have a small, but notable increase in flow turning and pressure ratio over the outer portion of the blade span. The data also suggest that the decrease in π(s) values near the outer span is related to loss generation mechanisms, and not due to incidence or exit flow angle changes.

The OL results are similar to the CH data for 0 < s < 0.4, albeit at slightly higher magnitudes. However, the τ(s) values increase from τ(s) ≈ 1 for s < 0.4 to τ(s) ≈ 1.07 at the tip. Only very small differences between the SW and CT data are observed. This further confirms that the casing slots were not influencing the compressor flow in any significant way at this OL operating point.

The SWNS temperature ratio data are again similar, with values near 1 at the hub, and increasing over the outer portion of the span with values generally higher than the OL condition. Interestingly, the τ(s) values for the SWNS conditions are nearly identical for SW and CT configurations. This is in contrast to the π(s) results that indicated an increase in pressure ratio with the CT. These data suggest that the flow turning is not greatly affected by the CT, but the loss mechanisms are. That is, the increases in the overall stage efficiency noted previously are a result of increases in pressure ratio (with flow turning essentially fixed) for 0.4 < s < 0.9.

Lastly, τ(s) results at CTNS condition show a similar trend as the SWNS values, but with a greater magnitude across the span. The maximum observed at the tip was approximately 1.14, demonstrating that significant increases in flow turning (compared with OL or SWNS) are possible with the CT geometry.

Additional detail and insight regarding the flow features associated with operating condition and casing configuration can be obtained by viewing the rotor-exit pressure and temperature ratios in the relative frame of reference. These are readily available from the numerical simulations, and the total pressure ratio data are shown in Figs. 14 and 15. In each image, a sector of the annulus is shown from hub to tip with a circumferential extent of approximately three times the blade pitch. The SW pressure ratio at the OL condition shows a distinct increase in pressure ratio near the hub (as anticipated from the mean profiles). The blade wakes appear to be narrow, with only a small region of high total pressure ratio near the tip that is assumed to be associated with the tip-leakage flow.

The pressure ratio for the CT configuration at the OL condition, shown in Fig. 15, shows nearly identical flow features compared with the SW results. This further confirms the understanding that the CT has essentially negligible influence on the rotor-passage flow at the OL condition.

The SW near-stall total pressure ratio in Fig. 14 shows a significant increase in pressure ratio at all locations, as expected. Additionally, the spatial extent of the region of very high pressure near the tip is greatly expanded compared with the OL condition. The CT results (at the CTNS condition) in Fig. 15 show similar results, with elevated pressure ratio across the entire domain. This further demonstrates that the effect of the CT is global, and not simply changing the tip-leakage flow.

The blade-level flow features were also investigated experimentally using time-resolved measurements of the rotor-exit total pressure. A “Kiel” type probe with an embedded miniature Kulite pressure transducer was traversed across the rotor-exit span. These data were collected at 98% speed for both SW and CT configurations at the four key conditions. The resulting data were phase-locked averaged to observe blade-to-blade flow features. The results were quantitatively and qualitatively similar to the numerical results presented in Figs. 14 and 15. In addition, the experimental values provide the opportunity to view the unsteadiness of the flow. This was evaluated by computing the root-mean-square (RMS) of the unsteady component of the exit total pressure. These values, normalized by mean total pressure ratio, are shown in Fig. 16. The three images show the results from the SWNS case (Fig. 16(a)), the CT configuration at the SWNS condition (Fig. 16(b)), and the CT configuration at the CTNS condition (Fig. 16(c)).

The SW configuration at NS indicates the highest levels of unsteadiness at the blade tip. The CT configuration with the same operating condition shows significantly reduced unsteadiness near the tip. Note that both the magnitude of the unsteadiness and the spatial extent of the region of high unsteadiness were reduced. Lastly, the CT configuration at the CTNS conditions shows the spatial extent of the tip flow unsteadiness to be even larger than the SW-NS condition, albeit the magnitude of the RMS values was slightly lower overall.

There were three main objectives of the work presented in this paper. The first was to provide quantitative measurements of the efficacy of a bent-skewed slot casing treatment. This was accomplished through a carefully designed experimental program. Significant efforts were made to ensure the highest possible build-to-build repeatability in the boundary conditions and the measurement systems. This was to ensure that changing between the SW and CT configurations resulted in changes in the measured performance, stall, and efficiency that were only a result of the change in casing.

The results indicated that the CT led to a substantial increase in pressure ratio and stall margin across all rotor speeds and operating conditions. The measurements also indicated an increase in adiabatic efficiency at all mass flowrates at the corrected design speed. At the nominal operating line, the increase was greater than 0.5%. This was considered to be an impactful result, given that the effectiveness of a CT for increasing stall margin is generally associated with a decrease in efficiency.

The second objective was to compare results from URANS simulations for both the SW and CT configurations to the experimental measurements. The results showed close agreement between the CFD and measured values of both the mean characteristic performance and the spanwise distributions of total pressure and total temperature.

The final objective was to use detailed measurements and numerical simulations to provide an improved understanding of the fluid mechanics that are altered by the casing slots. The experimental results demonstrated substantial changes in pressure ratio combined with small changes in flow turning across the entire span. The detailed measurements indicated that the CT greatly reduces the size and magnitude of the tip-leakage flow at both the OL and SWNS conditions.

This research was funded by The Office of Naval Research under agreement N000014-18-1-0246. Dr. Steven Martens was the program manager. Any opinions, findings, and conclusions or recommendations expressed in these materials are those of the author(s) and do not necessarily reflect the views of the Office of Naval Research.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$m˙c$ =

corrected mass flow

η =

stage adiabatic efficiency computed from system torque meter

π_r =

rotor total pressure ratio

π_s =

stage total pressure ratio

π(s) =

rotor pressure ratio as a function of span (s)

τ_r =

rotor total temperature ratio

τ_s =

stage total temperature ratio

τ_TM =

total temperature ratio computed from system torque meter

τ(s) =

rotor temperature ratio as a function of span (s)

SM =

stall margin of a given compressor characteristic