An improved prediction model for corner stall in axial compressors with dihedral effect

Jiabin LI, Lucheng JI

School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

KEYWORDS Axial compressor;Corner separation/stall prediction;Diffusion parameter;Dihedral effect;Three-dimensional blades

Abstract Corner stall predictions are important and difficult in axial compressors.However,all of the prediction models have proved to be ineffective for advanced compressor blades,which tend to use the combined sweep and dihedral. As for the prediction parameter DL, although it effectively modeled the effects of the adverse pressure gradient and secondary flow,it failed to predict the corner stall of curved blades because the model failed to consider the intersection of the boundary layer at the corner region.In this paper,the shape factor gradient Ψ of the boundary layer at the corner region was investigated by numerically studying specially shaped expansion pipes under different adverse pressure gradients. The improved prediction parameter DJ was presented based on the model of Ψ and the circumferential pressure gradient ξ. A comparison of the critical range of the prediction parameters DL and DJ was investigated using the NACA65 cascade database,which was established by a numerical method. Then, the stall criterion was validated according to the experimental results of various test facilities with different blade geometries and experimental conditions. The results show that the improved prediction parameter is able to predict the corner separation/stall flows and is in good agreement with the experimental results for axial compressors with three-dimensional designed blades.

1. Introduction

Endwall flow blockage and losses,especially three-dimensional flow separations at the corner, limit the aerodynamic loading levels of turbomachinery. Hence, over the past few decades,to accurately predict and avoid corner stall, many researches have been conducted to clarify the flow mechanisms and aerodynamic features of corner separation.1-6These studies demonstrate that the adverse pressure gradient and secondary cross flow are the key factors that affect corner stall. Therefore, prediction models for corner stall in axial compressors have been presented and have played a positive role in turbomachinery design.

According to an experimental study of a compressor cascade, De Haller7presented a diffusion parameter DH to predict corner stall, which is shown in Eq. (1). DH takes the

Notation AR Aspect ratio c Chord length Cp Adverse pressure gradient DF Lieblein diffusion factor DH De Haller diffusion factor DJ Ji diffusion factor DL Lei diffusion factor DY Yu diffusion factor H Shape factor h Span height i Incidence angle Ma Mach number PS Pressure surface pt Total pressure p Static pressure Re Reynolds number S Stall indicator SS Suction surface s Pitch of the blade passage W Mean streamwise velocity in blade passage Z Zweifel blade loading coefficient α Absolute flow angle β Expansion angle of the pipes γ Blade stagger angle δ Dihedral angle θ Blade camber angle λ Boundary layer momentum thickness σ Blade solidity Ψ Shape factor gradient Δ Boundary layer thickness Subscripts x Axial coordinates crit Critical value ss Suction surface ps Pressure surface 1 Inlet 2 Outlet

ratio of outlet to inlet velocity into account and is used to predict that corner stall occurs when DH<0.72.

Lieblein’s diffusion parameter DF takes the maximum velocity at the suction surface and exit velocity into account8and is used to predict that corner stall occurs when DF>0.6. The DF parameter is shown as

The diffusion parameters DH and DF are widely used in the compressor design. However, as the separation flows at the endwall corner are inherently three-dimensional, these two diffusion parameters are correlations that are obtained from a large amount of two-dimensional airfoil data; therefore, the prediction always fails near the endwall regions.

In view of this problem, Lei and Spakovsky9presented a diffusion parameter DL to predict endwall corner separation/stall flows during preliminary compressor design. The DL parameter takes into account both the adverse pressure gradient and secondary cross flow. The result indicates that corner stall occurs when DL>(0.4±0.05).The DL parameter is shown as

Yu and Liu10improved the DL parameter by further considering the blade aspect ratio and inflow boundary layer thickness. Diffusion parameter DY is presented as Eq. (4).The results show that the DY parameter accurately predicts corner stall and that corner stall occurs when DY>0.47±0.015.

Lei9and Yu10et al. reported that their parameters were ineffective for predicting the corner stall of advanced threedimension blades with a compound dihedral and sweep design. However, in the current state, three-dimensional blades are widely used in modern advanced compressor design. The practicability of using the above diffusion parameters for predicting corner stall is limited. It is important to propose a new diffusion parameter that can predict the corner stall of advanced three-dimensional blades more accurately.

It is noteworthy that both parameters DL and DY ignore the flow phenomenon,which is the boundary layer intersection at the corner region. In fact, the intersection of the boundary layers reduces the ability of the flow to resist the adverse pressure gradient, and often leads to the occurrence of the corner separation. Ji et al.11presented a model for describing the influence of the boundary layer intersection. From this model,the rules of the dihedral can be inferred, i.e. (A) corner flow becomes easier to separate when the dihedral angle is reduced,and vice versa; (B) the greater the 2nd derivatives of the dihedral angle are, the easier the flow is to separate. This paper investigates the development of the intersected boundary layer in specially shaped expansion pipes with different dihedral angles and expansion angles. The specially shaped expansion pipes are used as an equivalent model of the compressor passage. A new diffusion parameter is presented by taking into account the boundary layer intersection and secondary cross flow. The accuracy of the diffusion parameter for predicting corner stall is validated with numerical and experimental results.

2. Indicator of boundary layer separation

2.1. Indicator definition

The dihedral angle between the suction surface and endwall affects the development of the intersected boundary layer at the corner region. Ji et al.11presented a model to describe the influence of the boundary layer intersection. According to this model, when the boundary layer thicknesses of two intersecting solid walls are the same,the thickness of the intersected boundary layer at the corner can be described by Eq.(5),where λ is the boundary layer thickness at the solid surface.

The momentum loss thickness of the boundary layer satisfies the momentum equation of the boundary layer, which is shown as For incompressible flows, Eq. (6) can be transformed into Eq. (7) using the Bernoulli equation.

It is observed from Eq. (7) that the adverse pressure gradient and dihedral angle affect the shape factor of the intersected boundary layer.On the other hand,the boundary layer theory indicates that the boundary layer separates when the shape factor is larger than 2.2.12Hence, the dimensionless flow rate of the shape factor can be used to measure the degree of difficulty of boundary layer separation. The dimensionless flow rate of the shape factor is defined as Ψ, which is shown in Eq. (8).

2.2. Correlation of indicator Ψ

Experimental and numerical methods are usually adopted in studies of corner boundary layer development.13The correlation of indicator Ψ is investigated through numerical methods.The geometry of a compressor passage is too complex to be used to investigate the correlation of indicator Ψ. Specially shaped expansion pipes are used as an equivalent model of the compressor passage.The equivalence is reflected in the following two aspects:

(1) The dihedral angle of the specially shaped expansion pipes is the same as that of the compressor passage.The average of the dihedral angles at the inlet and outlet of the compressor passage is used if the dihedral angle varies along the flow passage.

(2) The adverse pressure gradient of the specially shaped expansion pipes is the same as that of the compressor passage. The adverse pressure gradient in the specially shaped expansion pipes is adjusted using different expansion angles.

Thus, the compressor passage can be modeled as specially shaped expansion pipes. An investigation of the development of the corner boundary layer can be conducted using specially shaped expansion pipes. Fig. 1 shows 4 types of specially shaped expansion pipes.The cross section of the pipes is a regular polygon. A flow analysis of the pipes was conducted in Ref.13.

Numerical studies are conducted for pipes with 6 different expansion angles. The hydraulic radius of these pipes is constant at 50 mm, and the inlet Mach numbers of these pipes are constant at 0.3. Further details of the numerical methods can be found in Ref.13. The numerical results indicate that the relationship among indicator Ψ, dihedral angle δ and expansion angle β (adverse pressure gradient Cp) is shown in Table 1. The adverse pressure gradient Cpis defined as

Table 1 indicates that indicator Ψ increases linearly with Cpwhen δ remains unchanged and decreases exponentially with δ when Cpremains unchanged.Thus,the correlation of indicator Ψ is assumed to be able to be calculated by Eq. (10). Fitting the data from Table 1, the coefficient in Eq. (10) can be obtained. The correlation of indicator Ψ can be determined by Eq. (11). The unit of the dihedral angle is degrees.

Fig.2 shows the correlation of indicator Ψ and the numerical data presented in Table 1, which are in good agreement with the dates. The R2values of the fitting results are larger than 0.9894.It should be noted that the Mach number has little effect in the subsonic region and Reynolds number has little effect in the self-modelling region.9Hence, Eq. (11) can be applied to all subsonic diffusion flows even though it is derived from the numerical results when the inlet Mach number is 0.3.Moreover, the compressor corner flows satisfy this condition.

Fig. 1 Specially shaped expansion pipes with different cross section shapes.

Table 2 Parameter space of linear cascade database.

4.3. Corner stall criterion with DL and DJ

To investigate the corner stall criterion with DL and DJ,a reliable stall indicator is necessary to detect the type of endwall corner flows during the modeling process.Lei and Spakovsky9presented a stall indicator S, as shown in Eq. (17). Lei and Spakovsky suggested a critical value of S of approximately 0.12.If S>0.12,corner stall occurs;otherwise,corner separation occurs. Indicator S is not always effective for measuring the severity degree of the corner separation in a small aspect ratio cascade.However,the aspect ratios of the cascades in this database are large, so indicator S can be effective.

Fig. 3 Grid dependence of simulation results.

Fig. 4 Prediction of endwall corner stall by diffusion parameter DL and stall criterion.

Fig. 5 Prediction of endwall corner stall by diffusion parameter DJ and stall criterion.

Figs. 4 and 5 show prediction of endwall corner stall using diffusion parameters DL and DJ at different incidence angles.Six cascades in the database show the occurrence of corner stall at 0°incidence.The diffusion parameter DL predicts corner stall when DL>0.41±0.04.The diffusion parameter DJ predicts corner stall when DJ>0.725±0.025. At +2° incidence,more cascades suffer corner stall.The diffusion parameters DL and DJ predict endwall corner stall when DL>0.43±0.01 and DJ>0.675±0.025. At+5° incidence, many cascades suffer corner stall and some cascades do not have a convergent numerical result. The diffusion parameters DL and DJ predict corner stall when DL>0.46 and DJ>0.765±0.035.The diffusion parameter DJ is more precise at 0° incidence, but worse at +2° incidence and +5°incidence.

Averaging the criteria of these three incidences, the criterion of corner stall with the DL parameter is DL>0.43±0.017,and the criterion of corner stall with the DJ parameter is DJ>0.725±0.028.

5. Validation of criterion with DJ

5.1. Validation with end-dihedral cascades

For modern three-dimensional compressor blades, the blade surface and endwall are often not perpendicular,and the dihedral angle between them varies along the flow direction. To verify the ability of the diffusion parameter DJ to predict corner stall, the end-dihedral cascade is designed as shown in Fig. 6. The first five sampling points in the database are selected to design the end-dihedral cascade. The dihedral angles of the cascades are set as 60°, 80°, 100°, and 120°.

Fig.7 shows the prediction results of endwall corner stall in end-dihedral cascades with diffusion parameters DL and DJ at 0° incidence. The stall indicator S decreases as δ increases,which indicates that a larger dihedral angle can improve corner separation. The most serious corner separation occurs when the dihedral angle is 60°.

The prediction results of endwall corner stall with the diffusion parameter DL are shown in Fig.7(a). The value of DL remains unchanged as the dihedral angle increases.As for cascade ④, DL fails to predict corner stall when the dihedral angle is 60°. Moreover, DL predicts that corner stall occurs in cascade ⑤when the dihedral angle is 120°, but corner separation occurs. Hence, diffusion parameter DL is completely invalid for predicting corner stall in 3D compressor blades.

The prediction results of endwall corner stall with diffusion parameter DJ are shown in Fig.7(b). The value of DJ decreases as the dihedral angle increases. Diffusion parameter DJ can reasonably predict the corner stall of the end-dihedral cascades.It is noteworthy that the stall indicator S increases in cascades ①and ②when the dihedral angle increases from 100° to 120°. This is mainly because the end-dihedral design can reduce the load level of the end section, which affects the value of the stall indicator S. The effect becomes more obvious when the corner separation is not serious.An independent stall parameter will lead to clearer results.

Fig. 6 Design parameters of end-dihedral cascade.

Fig. 7 Prediction results of endwall corner stall in end-dihedral cascades at 0° incidence.

It can be concluded that the diffusion parameter DJ has the ability to predict corner stall in an end-dihedral cascade,while diffusion parameter DL fails.However,for compressor blades,especially rotor blades, the dihedral angle in the end region often varies along the flow direction. The dihedral angle in parameter DJ is the average of the inlet and outlet dihedral angles. The ability of the diffusion parameter DJ to predict corner stall in compressor blades requires further analysis and experimental verification.

5.2. Experimental validation

To verify the ability of the diffusion parameter DJ to predict corner stall in three-dimensional axial-flow compressor blades,the geometric parameters of related blades were extensively extracted from published experimental research literature, as shown in Table 314-21The corresponding DL and DJ values are calculated as shown in Table 4. In Table 4,14-21‘N’ is for no corner stall,‘Y’ is for corner stall,and‘U’ is for uncertain.

The prediction results of the experimental validation cases with DL and DJ parameter are shown in Fig. 8. Prediction results with DJ parameter are more effective, especially for the curved blade, which are Cases 18, 26, 30, 31, 33, and 38.However, the prediction uncertainty of DJ parameter is slightly increased.

The corner stall criterion with the DL parameter is DL>0.43±0.017, and the corner stall criterion with the DJ parameter is DJ>0.725±0.028. According to Table 4,the diffusion parameter DJ can effectively predict corner stall in most cases. Based on these results, the DJ criterion is discussed below.

For the linear cascades that stack with NACA65 airfoil,the values of the DJ parameter in the literature are all below the criterion. It is predicted that no stall occurs in the corner region,which is verified by the experimental results.4,14,15Gbadebo also measured the flow field in a PVD cascade. The DJ parameter is 0.592,which is lower than the criterion.This prediction is in agreement with the experimental results.4Therefore, similar to the DL parameter, the DJ parameter can also effectively predict the corner stall of a linear cascade that stacks with the airfoil which is not NACA65.

The linear cascade experiment performed by Horlock et al.16was stacking with C4 airfoil.Corner stall was observed when the values of the DL parameter 0.33 and the DJ parameter 0.577 were subcritical.It is difficult to discuss the cause of the stall phenomenon in the corner region of the cascade with such limited information.

In a single stage compressor measured by Dong et al.17the C4 airfoil was adopted,and the blade was twisted.Corner stall was affected by the geometry of the near endwall sections,but not the geometry of the mid-span sections. The DJ parameter is calculated using the geometric parameters,which is the average of the three near endwall sections of the blade. The DJ parameters of the rotor and stator are 0.362 and 0.911,respectively, which are subcritical and supercritical. The DL parameter can also be used to obtain the same prediction results,which are in good agreement with the experimental results.

Schulz and Gallus2carried out an experimental study on a NACA65 annular cascade. The corner flow was measured under several inlet conditions. The experimental results showed that corner stall occurred at an inlet angle of 44.2°,but the results of the DL parameter and DJ parameter are 0.29 and 0.579,respectively,which are lower than the criterion.The reason is that the blade camber angle given by Schulz et al.was 29°, but the actual flow turning angle was approximately 60°. Lei reported that the blade camber angle should be 44.5°, and the results of the DL parameter and DJ parameter are 0.45 and 0.889,respectively.Thus,the parameter values are larger than the criterion.

Wellborn and Okiishi18experimentally measured the third stage stator flow field of a four-stage compressor. The stator was a three-dimensional blade,and the dihedral angle between the suction and the endwall varied from an acute angle to an obtuse angle along the flow direction.The experimental results showed that the stator suffered corner stall under the design condition. The average value of the inlet and outlet dihedral angle is used to calculate the DJ parameter. The values of DL parameter and DJ parameter are 0.420 and 0.815, respectively. The DL parameter predicts when corner stall may occur, while the DJ parameter predicts the actual corner stall.The prediction result of the DJ parameter is consistent with the experimental results.

Joslyn and Dring1studied second stage stator flow through a two-stage axial compressor test rig. The geometric parameters of the second stage stator were extracted as shown inTable 3.The value of the DL parameter and DJ parameter are 0.630 and 1.003, respectively, which are larger than the criterion. It is shown that both the DL parameter and DJ parameter predict the corner stall of the stator, which is consistent with the experimental measurements.

Table 3 Geometric parameters of compressor blades extracted from published experimental research literature.

Yocum and O’Brein19measured the flow field of a linear cascade at different incidences. The cascade airfoil was the same with the cascade used by Schulz and Gallus.2The results of the DL parameter and DJ parameter are shown in Table 4.For cases for which corner stall is going to occur, the DJ parameter successfully predicts that there is no corner stall,while the DL parameter predicts that the corner stall may occur. For cases for which corner stall has just occurred, the DL parameter successfully predicts its occurrence, while the prediction of the DJ parameter is uncertain.

Friedriches et al.20studied the effect of the threedimensional geometry on the flow field of a single-stage compressor stator. Two types of blades were used in the experiment. One is a standard blade stacking with NACA65-006 airfoil,and the other is a three-dimensional blade with forward swept leading edge. The experimental results show that corner stall occurred in the stator passage when standard blades were used, but no corner stall occurred when forward swept blades were used. The DL parameter predicts the occurrence of corner stall in standard blades, but fails to predict corner stall in the forward swept blade. The leading edge swept changed the dihedral angle at the endwall region,and the DL parameter does not have the ability to predict corner stall in a threedimensional blade. The DJ parameter is calculated to be 0.627,and the prediction is in agreement with the experimental results.

Zhong21carried out an experimental study on a compressor cascade with standard, positive lean, positive dihedral, negative dihedral and S-shaped cases under various operating conditions. The lean law of the five cascades is shown in Fig. 9,and the corresponding geometric parameters are shown in Table 3.For the standard case,prediction with the DL parameter is accurate,while prediction with the DJ parameter at 10°incidence is uncertain.For the positive dihedral case,the dihedral angle between the suction surface and the endwall is obtuse. The value of DL parameter remains unchanged with the standard case.The DJ parameter predicts that the endwall would not suffer corner stall anymore when the incidence is 10°, which is confirmed by the experimental results. For the positive lean case and the S-shaped case, the dihedral angle at the lower wall suction side corner region is the same as that of the positive dihedral case,and the prediction results with the DL parameter and DJ parameter also remain the same. It should be noted that the flow fields in the positive dihedral,positive lean,and S-shaped cases are totally different,and only the corner flow condition is the same. For the negative dihedral case, the dihedral angle between the suction surface and the endwall is acute.The DL parameter fails to predict corner stall when the incidence angles are 0°and 5°,while the prediction results with the DJ parameter are confirmed by the experimental results at all incidences.

It can be concluded that the DL parameter and DJ parameter have the same accuracy for predicting corner stall in linear blades.However,for three-dimensional blades,the DJ parameter can predict corner stall effectively,while the DL parameter fails. The DJ parameter still requires some improvements.First, the prediction accuracy needs to be improved. Second,the dihedral angle at the corner region always varies along the flow direction. Account for this problem will be an improvement direction in the future.

Table 4 Value of corresponding DL and DJ of compressor blades.

Fig. 8 Prediction results of experimental validation cases with DL and DJ parameter.

Fig. 9 Lean law of 5 experimental cascades.21

6. Conclusions

The prediction criterion of compressor corner stall can help to avoid the occurrence of corner stall in the compressor design.However, the existing diffusion parameters fail to predict the corner stall of 3D blades,which limit the practicability of these diffusion parameters.In this study,a new diffusion parameter,DJ, is modeled using basic geometric parameters. The DJ parameter takes into account the circumferential pressure gradient ξ and flow rate of the intersected boundary layer shape factor Ψ. The DJ parameter can be used to predict the corner stall of a 3D axial compressor blade.

Compared with the DL parameter, the DJ parameter also has the ability to predict the corner stall of a linear cascade.The database results show that the accuracy of the DJ parameter is higher than that of the DL parameter at 0° incidence,but lower at 2° and 10° incidences. Averaging the criterion of these three incidences, the corner stall criterion with the DL parameter is DL>0.43±0.017, and the corner stall criterion with the DJ parameter is DJ>0.725±0.028.

The diffusion parameter DJ has the ability to predict corner stall in an end-dihedral cascade,while diffusion parameter DL fails. The same conclusion can be drawn from the validation discussion based on the published experimental research literature.

Some problems should be addressed when the DJ parameter is used in 3D compressor blades.First,the DJ parameter is calculated using geometric parameters, which are the average of several near endwall sections of the blade.Second,the dihedral angle in the end region often varies along the flow direction. The dihedral angle in parameter DJ is the average of the inlet and outlet dihedral angles.

The DJ parameter still requires some improvements. First,the prediction accuracy needs to be improved by taking into account the aspect ratio and inflow boundary layer thickness.Second, the dihedral angle at the corner region always varies along the flow direction. These problems will be addressed in the future research.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 51676015).