- Email: [email protected]
Aerodynamics of swept and leaned transonic compressor-rotors
APPLIED ENERGY
Applied Energy 84 (2007) 1012–1027
www.elsevier.com/locate/apenergy
Aerodynamics of swept and leaned transonic compressor-rotors Ernesto Benini *, Roberto Biollo Department of Mechanical Engineering, University of Padova, Via Venezia, 1 – 35131 Padova, Italy Received 15 June 2006; received in revised form 23 February 2007; accepted 10 March 2007 Available online 12 June 2007
Abstract A systematic investigation to understand the impact of axially swept and tangentially leaned blades on the aerodynamic behaviour of transonic axial-flow compressor rotors was undertaken. Effects of axial and tangential blade curvature were analyzed separately. A commercial CFD package, which solves the Reynolds-averaged Navier–Stokes equations, was used to compute the complex flow field of transonic compressor-rotors. It was validated against NASA Rotor 37 existing experimental data. Computed performance maps and downstream profiles showed a good agreement with measured ones. Furthermore, comparisons with experimental data indicated that the overall features of three-dimensional shock structure, shock-boundary layer interaction, and wake development are calculated well by the numerical solution. Next, quite a large number of new transonic swept rotors (26) were modelled from the original Rotor 37, by changing the meridional curvature of the original stacking line through three previously defined control points (located at 33%, 67% and 100% of span). Similarly, 26 new transonic leaned rotors were modelled by changing the circumferential position of the same control points. All the new transonic rotors were simulated and the results revealed many interesting aspects which are believed to be very helpful to better understand the blade curvature effects on shock structure and secondary losses within a transonic rotor. 2007 Elsevier Ltd. All rights reserved. Keywords: Transonic compressor; Sweep; Lean; Shock waves; Blade shape
*
Corresponding author. Tel.: +39 049 8276767; fax: +39 049 8276785. E-mail address: [email protected] (E. Benini).
0306-2619/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.03.003
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1013
Nomenclature m_ p T
mass flow rate, kg/s pressure, Pa temperature, K
Subscripts choke choking condition 0 total 1 inlet 4 outlet
1. Introduction In modern high-performance aircraft gas-turbine engines, compressor stages must provide high values of both efficiency and compression ratio. This is important to minimize the fuel consumption and decrease the engine weight and size due to the reduction of number of stages. In response to these requirements, transonic axial-flow compressor rotors have been developed for many years. Due to the high blade-tip velocities, needed to obtain high pressure ratios, transonic compressor rotors develop intense shock-waves close to the blade tip and over part of the span. It is well known that the presence of the shocks and their interactions with other flow phenomena, such as tip clearance flows and boundary layers, induce high aerodynamic losses and entropy generation which negatively influence the overall efficiency. Actually, the application of sweep and lean on rotor blades is one of the most significant technological evolutions to improve the aerodynamic behaviour of transonic compressor stages. Different definitions of sweep and lean are used in the literature. Sweep and lean are related to the stacking of the blade sections, and the stacking may be defined on any two sets of perpendicular axes. A common choice is to define sweep and lean relative to the axial and tangential directions, so that sweep corresponds to moving the blade sections in the axial direction and lean to moving them in the tangential direction. Alternatively, sweep can be referred to the local chord-direction and lean to the corresponding orthogonal direction. The influence of sweep on shock structures and secondary flows has been widely analyzed in the literature and it seems to be of general agreement that the use of upstream swept blades can lead to significant benefits. Numerical and experimental analyses conducted by Hah et al. [1] to evaluate the performance of a conventional unswept rotor, a forward-swept rotor and an aft-swept rotor showed that the forward-swept rotor had a higher peak efficiency and a substantially larger stall margin than the baseline unswept rotor, and that the aft-swept rotor had a similar peak efficiency with a significantly smaller stall margin. Similar results were obtained by Wadia et al. [2] in a parallel investigation. Using a CFD model, on the other hand, Denton and Xu [3] observed that the global effects of swept blades on transonic fan-efficiency and pressure ratio are not very remarkable, but confirmed the significant improvements of compressor stability induced by the forward sweep.
1014
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
There are many aspects to be considered when trying to understand how sweep does influence the performance of a transonic blade-rotor. Three-dimensional shock structure is one of the most important. As described by Hah el al. [1], the shock must intersect the casing at right angles (a phenomenon known as ‘‘endwall effect’’); this fact induces the shock to move upstream in an aft-swept rotor and downstream in a forward-swept rotor. Usually, a shock which is located more downstream near the casing leads to a better stability. In addition, lower down the blade, where the endwall effect has no influence, sweep can induce major effects on the shock pattern; it is known that, shock sweep can reduce the shock losses and hence improve the local efficiency. Sweep has also a considerable influence on the accumulation of low momentum fluid near the endwall tip region, which is believed to have a very negative effect on rotor stability. Yamaguchi et al. [4] found that, for a forward-swept rotors, this phenomenon is of lesser importance than in conventional radial rotor blades due to the decreased radial migration of fluid particles within the suction-side boundary-layer (a secondary flow that follows the imbalance between the centrifugal force and the pressure gradient). Moreover, as observed by Denton and Xu [5], sweep influences the loading on the blade near the walls; in particular, forward sweep can reduce the blade loading in the front area of the tip region, where the loading rapidly falls to zero (no blade) as one moves radially from the tip to a lower span. This helps reduce the sensitivity to changes in incidence and the intensity of the tip leakage flows in this area. The influence of blade lean in transonic compressor rotors is not extensively described in the literature, but it seems, as numerically observed by Bergner et al. [6], that the use of lean can give rise to a significant change in the shock pattern. Using CFD, Ahn and Kim [7] analyzed the impact of lean when the blade is skewed towards the direction of rotation and observed a positive influence on the overall rotor-efficiency. Recently, Benini [8] performed a multi-objective design optimization on the NASA Rotor 37 and demonstrated that the overall efficiency can be significantly improved by giving the blade a proper lean towards the direction of rotation, thanks to a drastic modification in the shock structure within the blade passage. In the present paper, a systematic numerical investigation has been conducted to better understand the blade curvature effects on the complex aerodynamic behaviour of a transonic compressor rotor. Lean and sweep were defined here relative to the axial and tangential directions. So, quite a large number of transonic axially-swept and tangentially-leaned rotors were modelled and successively analyzed using an accurate CFD model. 2. Investigated rotor geometries The Rotor 37, a well known test case design by NASA Lewis Research Center and representative of complex three-dimensional viscous flow structures in transonic bladings, was selected as a baseline radially-stacked rotor from which new swept and leaned rotors were derived. Some design information and overall stage performance came from Reid and Moore [9], and more detailed measurement data were provided by Moore and Reid [10]. Fig. 1 shows the meridional view of the rotor. The rotor had 36 Multiple-Circular-Arc (MCA) blades with an inlet hub-tip diameter ratio of 0.7, a blade aspect ratio of 1.19 and a tip solidity of 1.29. The running tipclearance was 0.0356 cm (0.45% of the blade span). Design pressure ratio and adiabatic efficiency were respectively 2.106 and 0.877 at a mass flow of 20.19 kg/s. The rotor had
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1015
Fig. 1. Meridional view of NASA Rotor 37 – sketch.
a measured choking mass flow of 20.93 kg/s. The design wheel speed was 17188 rpm, with a nominal tip speed of 454 m/s. The new swept and leaned rotors were modelled by changing the curvature of the original blade stacking line of Rotor 37 in the meridional and circumferential planes, respectively. As schematically shown in Fig. 2, the radial stacking line of Rotor 37 (the solid vertical line) was modified by moving the three control-points located on 33%, 67% and 100% span from the hub (signed as black circles) on the corresponding predefined positions (the ·-marked locations). All possible combinations were considered (26 for sweep CIRCUMFERENTIAL PLANE Lean
MERIDIONAL PLANE Sweep Upstream Downstream 100
100
C
80
60
Control points can be moved on corresponding xmarked locations
40
A
Span %
Span %
80
Rotor 37 stacking line
B New leaned stacking lines
60
40
20
20
D
New swept stacking lines 0 -0.8
Direction of rotation
0
-0.4
0
0.4
Axial displacement [cm]
0.8
-3
-1
1
Tangential displacement [deg]
Fig. 2. Swept and leaned-stacking line definition.
3
1016
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Fig. 3(a). Application of swept stacking line ‘‘A’’ of Fig. 2.
Fig. 3(b). Application of leaned stacking line ‘‘B’’ of Fig. 2.
and 26 for lean). Dashed lines represent some of all new stacking lines so obtained. Figs. 3(a) and 3(b) show, for instance, the blades obtained using the stacking lines designated as ‘‘A’’ and ‘‘B’’ in Fig. 2. Being the tip endwall region characterized by very complex flow structures, it was preferred not to change the meridional position of the tip-blade profile, with the aim of avoiding any other variables of influence. Thus, the curvature of the new swept stacking lines was applied to the blade starting always from the tip section instead of from the hub section, as shown in Fig. 4 for the swept stacking line ‘‘C’’ of Fig. 2. Here, the adjective of ‘‘forward’’ was given to lean when the blade results skewed toward the direction of rotation and to sweep when the blade results curved upstream. In this work, no attempt was made to re-design the blades for each stack option to retain the passage area distribution qualities of Rotor 37 in terms of throat margin, start margin, and effective camber [2,11]; thus, the perceived performance changes with the new rotors could not be a consequence of blades curvature only. 3. Flow solver: description and validation For all the investigated rotor geometries, the flow field around the blade was computed using the commercial CFD code CFX 5.7.1 [12], where the 3D Reynolds-averaged form of the Navier–Stokes equations are solved using a finite-element based finite-volume method. An Algebraic Multigrid method based on the Additive Correction Multigrid (ACM) strategy [13] was used. In order to increase the accuracy of results, a high-resolution advection scheme was adopted to calculate the advection terms in the discrete finite-volume
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1017
MERIDIONAL PLANE Sweep Upstream Downstream 100
Span %
80
Curvature application
60
Curvature definition 40
20
0 -0.8
-0.4
0
0.4
0.8
Axial displacement [cm]
Fig. 4. The curvature of new swept stacking lines was applied to the original blade maintaining the axial position of the tip profile.
equations. Steady state solutions were computed using the k–e turbulence model [14] along with scalable wall functions. A multi-block structured grid (Fig. 5(a)), of about 500,000 cells, was adopted to discretize the computational domain. An H-type grid was used for both the inlet and outlet blocks, while a composite J/O-grid was utilized for the passage block in order to reduce grid skewness and to facilitate solving the boundary layer around the blade (Fig. 5(b)). The tip-clearance gap was also modelled and the relative motion between rotor and casing was included.
Fig. 5(a). Example of multi-block grid adopted – computational domain (CFX-TurboGrid 2.2 [15]).
1018
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
The total pressure, total temperature and flow angle were fixed at the inflow boundary, while the average static-pressure was imposed at the outflow. Periodic boundary conditions were applied on the lateral faces of the flow domain. The walls were treated as smooth and adiabatic. An angular velocity corresponding to the nominal rotational-speed of Rotor 37 was applied. Subsequent to the tests of Moore and Reid [10], the Rotor 37 was retested as an isolated component. The test facility was described by Suder et al. [16,17] and the measurements, used for the AGARD test case for turbomachinery CFD codes [18] were utilized in this work for the model validation. Fig. 6 shows the axial location of some measurement stations. Flow surveys were undertaken for stations 1 and 4, which correspond to the inlet and outlet boundaries of the numerical model. In order to reproduce the test boundary
Fig. 5(b). Example of multi-block grid adopted – hub view (CFX-TurboGrid 2.2 [15]).
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1019
Fig. 6. Measurement stations within NASA Rotor 37.
conditions, the inlet total pressure and total temperature were fixed at p01 = 101,325 Pa and T01 = 288.15 K, respectively. Measured and calculated rotor performance maps are presented in Fig. 7(a). The experimental and computed mass-flow rates were normalized using the corresponding
Total pressure ratio
2.20 2.15 2.10 2.05 2.00
Experiment CFD
1.95 1.90 0.90
0.92
0.94 0.96 Normalized mass flow
0.98
1.00
0.98
1.00
Adiabatic efficiency
0.900 0.880 0.860 0.840 0.820
Experiment CFD
0.800 0.780 0.90
0.92
0.94
0.96
Normalized mass flow
Fig. 7(a). Performance maps. Experimental data from AGARD [18].
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Span %
1020 100
100
100
80
80
80
60
60
60
40
40
40
20
20
CFD
20
Exp
0
0 1.7
1.9
2.1
Total pressure ratio
2.3
0 1.1
1.2
1.3
Total temperature ratio
1.4
0.6
0.7
0.8
0.9
1
Adiabatic efficiency
_ m_ choke ¼ 98%. Experimental data from AGARD [18]. Fig. 7(b). Radial plots at station 4: m=
mass-flow at choking condition. The model predicted well the choking mass-flow, giving a value of 20.96 kg/s against the measured one of 20.93 kg/s. The efficiency is slightly underestimated in the peak zone (by about 2%), while the calculated total pressure ratio p04/p01 is almost within the measurement uncertainties in all the operating conditions. Fig. 7(b) shows the spanwise distributions of pitch-averaged total pressure, total temperature and adiabatic efficiency at the outlet for 98% of choking mass flow; calculated profiles are in good agreement with the measurement data. Three-dimensional shock structure and secondary flows (tip-clearance flows, blade wakes, etc.) were also well predicted by the model. Comparisons with experimental data can be found in a recent work of the authors [19]. 4. Results All the modelled rotors were simulated, fixing the same static pressure at the outlet of the computational domain. Even if this is not the appropriate boundary condition for consistently comparing the performance of different designs (each rotor can be at a different aerodynamic-loading level), this approach makes possible a direct comparison with the baseline performance. Fig. 8 compares the predicted efficiency of each new rotor to the computed one of Rotor 37 at the same normalized mass-flow. It is clear that no information can be obtained in this way about the complete performance maps. For each rotor, the mass-flow rate was normalized using the corresponding computed choking mass-flow. For completeness, Fig. 9 shows the predicted choking mass flow rate of different designs. The rotors named as ‘‘others’’ in Figs. 8 and 9 correspond to those modelled rotors which cannot simply be classified as ‘‘forward or backward curved’’, such as the rotor derived from the application of stacking line ‘‘D’’ of Fig. 2. Not very high increments in the overall efficiency were induced by the introduction of a swept stacking line. In Fig. 8(a), the rotors with a leading-edge profile regularly curved downstream (backward swept) are marked with triangles, while the rhombuses indicate all the rotors with a leading edge significantly turned upstream (forward swept). Best
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1021
0.880
Adiabatic efficiency
Rotor 37 0.870
Forward-swept rotors Aft-swept rotors
0.860
Others
0.850
0.840
0.830 0.90
0.92
0.94
0.96
098 .
100 .
0.98
1.00
Normalized mass flow
Swept modelled rotors
Rotor 37
0.870
Forward leaned rotors Backward leaned rotors
0.860
Others
0.850
0.840
0.830 0.90
0.92
0.94
0.96
Normalized mass flow
Leaned modelled rotors Fig. 8. Computed adiabatic efficiency.
21.5 21.3
Backward leaned
21.1
Rotor 37
Others
Aft-swept
21.7
20.9 20.7 20.5 20.3 20.1
Forward swept
Choking flow rate[kg/s]
Adiabatic efficiency
0.880
Forward leaned
19.9
Fig. 9. Choking mass-flow rate of different designs.
1022
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027 MERIDIONAL PLANE
CIRCUMFERENTIAL PLANE
Downstream
100
100
80
80
Span %
Span %
Upstream
60
60
40
40
20
20
0 -0.8
0 -0.4
0
0.4
Axial displacement [cm]
0.8
-3
-1
1
3
Tangential displacement [deg]
Swept rotor
Leaned rotor
Fig. 10. Rotors described.
results were obtained by the backward swept rotors, which achieved an efficiency of about 0.6% higher than that of Rotor 37 referred to the same operating condition. The same rotors have also produced the highest choking mass-flows, with a maximum of about 4% higher compared with the baseline rotor. Similar results were obtained by Abdelhamid et al. [20]. The application of forward sweep, instead, did not provide substantial improvements in the overall efficiency and also resulted in a lower choking mass-flow. As shown in Fig. 8(b), a considerably better performance was achieved using forward leaned rotor-blades, with an efficiency increase of 1.2% above that of Rotor 37. On the other hand, as expected, the backward-leaned curvature was detrimental in most cases. It was also observed that all the leaned rotors produced a very similar choking mass-flow to that of Rotor 37 (Fig. 9). In the following discussion, the backward-swept rotor and the forward-leaned rotor defined in Fig. 10 are analyzed with respect to the original Rotor 37. In Fig. 11, the computed performance maps are compared. Substantial improvements in the adiabatic efficiency were obtained in all operating conditions for both swept and leaned rotors (approximately 1.3% higher than that of Rotor 37 at 99% choking mass flow for the leaned rotor). Moreover, a slightly higher total-pressure ratio was achieved over a large part of the operating range. As shown, the three rotors denote approximately the same stall margin. The near-stall operating point was computed as the last point for which the steady model implemented was able to converge. Fig. 12 compares the computed radial distributions of performance quantities at the peak efficiency condition (close to 99% choking mass flow). At the outer span, Rotor 37 gave a slightly superior efficiency due exclusively to a higher total-pressure ratio delivered. On the other hand, between 30% and 80% span height, for both swept and leaned rotors, a substantially better performance can be observed. This is a direct consequence of the three-dimensional shock structure induced by the blade curvature; in this region which
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1023
2.20
Total pressure ratio
2.15 2.10 2.05 R37 2.00
Leaned rotor Swept rotor
1.95 1.90 0.90
0.92
0.94
0.96
0.98
1.00
Normalized mass flow
Adiabatic efficiency
0.880
0.870
0.860
0.850
0.840
0.830 0.90
0.92
0.94
0.96
0.98
1.00
Normalized mass flow
Fig. 11. Performance maps.
100
100
80
80
80
60
60
60
40
40
20
20
Span %
100
40 R37
20
0 1.85
Sw Ln
1.95 2.05
2.15
Total pressure ratio
2.25
0 1.22
0 1.24
1.26
1.28
1.3
Total temperature ratio
0.7
0.8
0.9
1
Adiabatic efficiency
Fig. 12. Radial plots at outlet – peak efficiency.
differs from the tip region, where there are many flow phenomena which interact with each other leading to a complex flow field, the shock is the flow phenomenon which can mainly influence the local performance of the rotor.
1024
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Fig. 13 shows the peak-efficiency distribution for the relative Mach number (Mach number distribution relative to the rotor reference frame) inside the blade passage. As apparent, the spatial shape of the shock is clearly influenced by the blade curvature. At the upper half of the span height, the leaned rotor presents a shock front considerably
Fig. 13. Comparison of relative Mach-number distributions – peak efficiency.
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1025
curved downstream in the blade-to-blade view (Fig. 13(b) and (c)); consequently, a slightly aft-swept inclination is also induced in the meridional view (Fig. 13(a)). This certainly helps to reduce aerodynamic shock losses, leading to positive effects on the overall efficiency; on the other hand, the shock interacts with the suction surface more downstream, giving rise to the risk of inducing boundary-layer separation towards the rear. For the swept rotor, the shock presents a more oblique pattern from hub to tip compared with
Fig. 14. Comparison of relative Mach-number distributions – near stall.
1026
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
the unswept rotor, as a consequence of the meridional blade leading-edge curvature; no substantial changes are noticeable in the blade-to-blade views with respect to the Rotor 37. The relative Mach number distribution, at near stall operating condition, is shown in Fig. 14. As expected, all the rotors produce a shock front detached from the blade leading-edge and located further upstream compared with the shock at the corresponding peak-efficiency condition, as consequence of the higher incidence angle due to the lower mass-flow rate. It is interesting to point out that, while the Rotor 37 and the backwardswept rotor gave rise to a similar normal shock situation at both 70% and 90% span, the forward-leaned rotor showed a slightly less detached shock front at 90% span. 5. Conclusions A systematic analysis in order to understand the effects of swept and leaned blades on the aerodynamics of transonic axial-flow compressor rotors was carried out. Many new transonic axially-swept and tangentially-leaned rotors were modelled from the radiallystacked NASA Rotor 37, by changing respectively the meridional and tangential curvatures of the original stacking line. Sweep and lean effects were separately analyzed. An attempt was made to not modify any other design parameter. A commercial CFD code, which solves the Reynolds-averaged Navier–Stokes equations, was successfully validated and used to solve the complex flow-field inside the rotors. The backward swept rotors had a slightly better performance compared with the forward-swept rotors, with a predicted efficiency of about 0.5–0.6% higher than that of Rotor 37 under the same operating-conditions; in addition, a significant higher choking mass-flow was observed. The specific backward-swept rotor analyzed showed the possibility of obtaining a considerable aft-swept shock front with a convenient meridional leadingedge curvature. As expected, a substantial increment in the overall efficiency (up to 1.3% higher than that of Rotor 37) was obtained by applying a stacking line curved toward the direction of rotation (forward lean). A shock front visibly curved downstream on the blade-to-blade plane was observed in all the forward-leaned rotors modelled, with a considerable reduction of shock strength. Very little influence was observed on the choking mass-flow rate. The three-dimensional shock structure represents one of the most important aspects to take into account in order to reduce shock losses and to improve the aerodynamic behaviour of the rotor; it was demonstrated that the use of swept or leaned blades can effectively help to achieve this aim. Further improvements in the overall efficiency can be expected from the simultaneous use of sweep and lean. References [1] Hah C, Puterbaugh SL, Wadia AR. Control of shock structure and secondary-flow field inside transonic compressor rotors through aerodynamic sweep, ASME Paper 98-GT-561, 1998. [2] Wadia AR, Szucs PN, Crall DW. Inner workings of aerodynamic sweep. ASME J Turbomachinery 1998;120(4):671–82. [3] Denton JD, Xu L. The effects of lean and sweep on transonic fan performance, ASME Paper GT-200230327, 2002. [4] Yamaguchi N, Tominaga T, Hattori S, Mitsubishi T. Secondary-loss reduction by forward-skewing of axial compressor rotor blading. In: Proceedings of 1991 Yokohama international gas turbine congress, vol. 2, 1991.
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1027
[5] Denton JD, Xu L. The exploitation of three-dimensional flow in turbomachinery design, VKI lecture series 1999–2002 – turbomachinery blade design system, Belgium, 1999. [6] Bergner J, Hennecke DK, Hoeger M, Engel K. Darmstadt Rotor No. 2 – part II design of lean rotor blades, ISROMAC 9, Honolulu, Hawaii, USA, February 10–14, 2002. [7] Ahn CS, Kim KY. Aerodynamic-design optimization of an axial compressor rotor, ASME Paper GT-200230445, 2002. [8] Benini E. Three-dimensional multi-objective design optimization of a transonic compressor rotor. AIAA J Propul Power 2004;20(3):559–65. [9] Reid L, Moore RD. Design and overall performance of four highly-loaded, high-speed inlet stages for an advanced high-pressure-ratio core compressor, NASA TP 1337, 1978. [10] Moore RD, Reid L. Performance of single-stage axial flow transonic compressor with rotor and stator aspect-ratios of 1.19 and 1.26, respectively, and with a design pressure ratio of 2.05, NASA TP 1659, 1980. [11] Wadia AR, Copenhaver WW. An investigation of the effect of cascade area ratios on transonic compressor performance. ASME J Turbomachinery 1996;118(4):760–70. [12] CFX 5.7.1, ANSYS, Inc., Canonsburg, PA 15317, USA. [13] Hutchinson BR, Raithby GD. A multigrid method based on additive correction strategy. Numer HeatTransfer 1986;9:511–37. [14] Launder B, Spalding D. The numerical computation of turbulent flow. Comp Math Appl Mech Eng 1974;3:269–89. [15] CFX TurboGrid 2.2, ANSYS, Inc., Canonsburg, PA 15317, USA. [16] Suder KL, Celestina ML. Experimental and computational investigation of the tip-clearance flow in a transonic axial-compressor rotor. ASME J Turbomachinery 1996;118(2):218–29. [17] Suder KL, Chima RV, Strazisar AJ, Roberts WB. The effect of adding roughness and thickness to a transonic axial-compressor rotor. ASME J Turbomachinery 1995;117(4):491–505. [18] AGARD, CFD validation for propulsion system components, Agard-AR-355, May 1998. [19] Biollo R, Benini E. Validation of a Navier–Stokes solver for CFD computations of transonic compressors, ESDA2006-95318, 2006. [20] Abdelhamid HF, Shreeve RP, Hobson GV. Sweep in a transonic fan-rotor: part 2. CFD and stress analyses. Presented at the 43rd ASME gas-turbine and aeroengine technical congress, exposition and user’s symposium, June 2–5, 1998, Stockholm, Sweden, ASME Paper 98-GU-579.
Applied Energy 84 (2007) 1012–1027
www.elsevier.com/locate/apenergy
Aerodynamics of swept and leaned transonic compressor-rotors Ernesto Benini *, Roberto Biollo Department of Mechanical Engineering, University of Padova, Via Venezia, 1 – 35131 Padova, Italy Received 15 June 2006; received in revised form 23 February 2007; accepted 10 March 2007 Available online 12 June 2007
Abstract A systematic investigation to understand the impact of axially swept and tangentially leaned blades on the aerodynamic behaviour of transonic axial-flow compressor rotors was undertaken. Effects of axial and tangential blade curvature were analyzed separately. A commercial CFD package, which solves the Reynolds-averaged Navier–Stokes equations, was used to compute the complex flow field of transonic compressor-rotors. It was validated against NASA Rotor 37 existing experimental data. Computed performance maps and downstream profiles showed a good agreement with measured ones. Furthermore, comparisons with experimental data indicated that the overall features of three-dimensional shock structure, shock-boundary layer interaction, and wake development are calculated well by the numerical solution. Next, quite a large number of new transonic swept rotors (26) were modelled from the original Rotor 37, by changing the meridional curvature of the original stacking line through three previously defined control points (located at 33%, 67% and 100% of span). Similarly, 26 new transonic leaned rotors were modelled by changing the circumferential position of the same control points. All the new transonic rotors were simulated and the results revealed many interesting aspects which are believed to be very helpful to better understand the blade curvature effects on shock structure and secondary losses within a transonic rotor. 2007 Elsevier Ltd. All rights reserved. Keywords: Transonic compressor; Sweep; Lean; Shock waves; Blade shape
*
Corresponding author. Tel.: +39 049 8276767; fax: +39 049 8276785. E-mail address: [email protected] (E. Benini).
0306-2619/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.03.003
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1013
Nomenclature m_ p T
mass flow rate, kg/s pressure, Pa temperature, K
Subscripts choke choking condition 0 total 1 inlet 4 outlet
1. Introduction In modern high-performance aircraft gas-turbine engines, compressor stages must provide high values of both efficiency and compression ratio. This is important to minimize the fuel consumption and decrease the engine weight and size due to the reduction of number of stages. In response to these requirements, transonic axial-flow compressor rotors have been developed for many years. Due to the high blade-tip velocities, needed to obtain high pressure ratios, transonic compressor rotors develop intense shock-waves close to the blade tip and over part of the span. It is well known that the presence of the shocks and their interactions with other flow phenomena, such as tip clearance flows and boundary layers, induce high aerodynamic losses and entropy generation which negatively influence the overall efficiency. Actually, the application of sweep and lean on rotor blades is one of the most significant technological evolutions to improve the aerodynamic behaviour of transonic compressor stages. Different definitions of sweep and lean are used in the literature. Sweep and lean are related to the stacking of the blade sections, and the stacking may be defined on any two sets of perpendicular axes. A common choice is to define sweep and lean relative to the axial and tangential directions, so that sweep corresponds to moving the blade sections in the axial direction and lean to moving them in the tangential direction. Alternatively, sweep can be referred to the local chord-direction and lean to the corresponding orthogonal direction. The influence of sweep on shock structures and secondary flows has been widely analyzed in the literature and it seems to be of general agreement that the use of upstream swept blades can lead to significant benefits. Numerical and experimental analyses conducted by Hah et al. [1] to evaluate the performance of a conventional unswept rotor, a forward-swept rotor and an aft-swept rotor showed that the forward-swept rotor had a higher peak efficiency and a substantially larger stall margin than the baseline unswept rotor, and that the aft-swept rotor had a similar peak efficiency with a significantly smaller stall margin. Similar results were obtained by Wadia et al. [2] in a parallel investigation. Using a CFD model, on the other hand, Denton and Xu [3] observed that the global effects of swept blades on transonic fan-efficiency and pressure ratio are not very remarkable, but confirmed the significant improvements of compressor stability induced by the forward sweep.
1014
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
There are many aspects to be considered when trying to understand how sweep does influence the performance of a transonic blade-rotor. Three-dimensional shock structure is one of the most important. As described by Hah el al. [1], the shock must intersect the casing at right angles (a phenomenon known as ‘‘endwall effect’’); this fact induces the shock to move upstream in an aft-swept rotor and downstream in a forward-swept rotor. Usually, a shock which is located more downstream near the casing leads to a better stability. In addition, lower down the blade, where the endwall effect has no influence, sweep can induce major effects on the shock pattern; it is known that, shock sweep can reduce the shock losses and hence improve the local efficiency. Sweep has also a considerable influence on the accumulation of low momentum fluid near the endwall tip region, which is believed to have a very negative effect on rotor stability. Yamaguchi et al. [4] found that, for a forward-swept rotors, this phenomenon is of lesser importance than in conventional radial rotor blades due to the decreased radial migration of fluid particles within the suction-side boundary-layer (a secondary flow that follows the imbalance between the centrifugal force and the pressure gradient). Moreover, as observed by Denton and Xu [5], sweep influences the loading on the blade near the walls; in particular, forward sweep can reduce the blade loading in the front area of the tip region, where the loading rapidly falls to zero (no blade) as one moves radially from the tip to a lower span. This helps reduce the sensitivity to changes in incidence and the intensity of the tip leakage flows in this area. The influence of blade lean in transonic compressor rotors is not extensively described in the literature, but it seems, as numerically observed by Bergner et al. [6], that the use of lean can give rise to a significant change in the shock pattern. Using CFD, Ahn and Kim [7] analyzed the impact of lean when the blade is skewed towards the direction of rotation and observed a positive influence on the overall rotor-efficiency. Recently, Benini [8] performed a multi-objective design optimization on the NASA Rotor 37 and demonstrated that the overall efficiency can be significantly improved by giving the blade a proper lean towards the direction of rotation, thanks to a drastic modification in the shock structure within the blade passage. In the present paper, a systematic numerical investigation has been conducted to better understand the blade curvature effects on the complex aerodynamic behaviour of a transonic compressor rotor. Lean and sweep were defined here relative to the axial and tangential directions. So, quite a large number of transonic axially-swept and tangentially-leaned rotors were modelled and successively analyzed using an accurate CFD model. 2. Investigated rotor geometries The Rotor 37, a well known test case design by NASA Lewis Research Center and representative of complex three-dimensional viscous flow structures in transonic bladings, was selected as a baseline radially-stacked rotor from which new swept and leaned rotors were derived. Some design information and overall stage performance came from Reid and Moore [9], and more detailed measurement data were provided by Moore and Reid [10]. Fig. 1 shows the meridional view of the rotor. The rotor had 36 Multiple-Circular-Arc (MCA) blades with an inlet hub-tip diameter ratio of 0.7, a blade aspect ratio of 1.19 and a tip solidity of 1.29. The running tipclearance was 0.0356 cm (0.45% of the blade span). Design pressure ratio and adiabatic efficiency were respectively 2.106 and 0.877 at a mass flow of 20.19 kg/s. The rotor had
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1015
Fig. 1. Meridional view of NASA Rotor 37 – sketch.
a measured choking mass flow of 20.93 kg/s. The design wheel speed was 17188 rpm, with a nominal tip speed of 454 m/s. The new swept and leaned rotors were modelled by changing the curvature of the original blade stacking line of Rotor 37 in the meridional and circumferential planes, respectively. As schematically shown in Fig. 2, the radial stacking line of Rotor 37 (the solid vertical line) was modified by moving the three control-points located on 33%, 67% and 100% span from the hub (signed as black circles) on the corresponding predefined positions (the ·-marked locations). All possible combinations were considered (26 for sweep CIRCUMFERENTIAL PLANE Lean
MERIDIONAL PLANE Sweep Upstream Downstream 100
100
C
80
60
Control points can be moved on corresponding xmarked locations
40
A
Span %
Span %
80
Rotor 37 stacking line
B New leaned stacking lines
60
40
20
20
D
New swept stacking lines 0 -0.8
Direction of rotation
0
-0.4
0
0.4
Axial displacement [cm]
0.8
-3
-1
1
Tangential displacement [deg]
Fig. 2. Swept and leaned-stacking line definition.
3
1016
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Fig. 3(a). Application of swept stacking line ‘‘A’’ of Fig. 2.
Fig. 3(b). Application of leaned stacking line ‘‘B’’ of Fig. 2.
and 26 for lean). Dashed lines represent some of all new stacking lines so obtained. Figs. 3(a) and 3(b) show, for instance, the blades obtained using the stacking lines designated as ‘‘A’’ and ‘‘B’’ in Fig. 2. Being the tip endwall region characterized by very complex flow structures, it was preferred not to change the meridional position of the tip-blade profile, with the aim of avoiding any other variables of influence. Thus, the curvature of the new swept stacking lines was applied to the blade starting always from the tip section instead of from the hub section, as shown in Fig. 4 for the swept stacking line ‘‘C’’ of Fig. 2. Here, the adjective of ‘‘forward’’ was given to lean when the blade results skewed toward the direction of rotation and to sweep when the blade results curved upstream. In this work, no attempt was made to re-design the blades for each stack option to retain the passage area distribution qualities of Rotor 37 in terms of throat margin, start margin, and effective camber [2,11]; thus, the perceived performance changes with the new rotors could not be a consequence of blades curvature only. 3. Flow solver: description and validation For all the investigated rotor geometries, the flow field around the blade was computed using the commercial CFD code CFX 5.7.1 [12], where the 3D Reynolds-averaged form of the Navier–Stokes equations are solved using a finite-element based finite-volume method. An Algebraic Multigrid method based on the Additive Correction Multigrid (ACM) strategy [13] was used. In order to increase the accuracy of results, a high-resolution advection scheme was adopted to calculate the advection terms in the discrete finite-volume
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1017
MERIDIONAL PLANE Sweep Upstream Downstream 100
Span %
80
Curvature application
60
Curvature definition 40
20
0 -0.8
-0.4
0
0.4
0.8
Axial displacement [cm]
Fig. 4. The curvature of new swept stacking lines was applied to the original blade maintaining the axial position of the tip profile.
equations. Steady state solutions were computed using the k–e turbulence model [14] along with scalable wall functions. A multi-block structured grid (Fig. 5(a)), of about 500,000 cells, was adopted to discretize the computational domain. An H-type grid was used for both the inlet and outlet blocks, while a composite J/O-grid was utilized for the passage block in order to reduce grid skewness and to facilitate solving the boundary layer around the blade (Fig. 5(b)). The tip-clearance gap was also modelled and the relative motion between rotor and casing was included.
Fig. 5(a). Example of multi-block grid adopted – computational domain (CFX-TurboGrid 2.2 [15]).
1018
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
The total pressure, total temperature and flow angle were fixed at the inflow boundary, while the average static-pressure was imposed at the outflow. Periodic boundary conditions were applied on the lateral faces of the flow domain. The walls were treated as smooth and adiabatic. An angular velocity corresponding to the nominal rotational-speed of Rotor 37 was applied. Subsequent to the tests of Moore and Reid [10], the Rotor 37 was retested as an isolated component. The test facility was described by Suder et al. [16,17] and the measurements, used for the AGARD test case for turbomachinery CFD codes [18] were utilized in this work for the model validation. Fig. 6 shows the axial location of some measurement stations. Flow surveys were undertaken for stations 1 and 4, which correspond to the inlet and outlet boundaries of the numerical model. In order to reproduce the test boundary
Fig. 5(b). Example of multi-block grid adopted – hub view (CFX-TurboGrid 2.2 [15]).
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1019
Fig. 6. Measurement stations within NASA Rotor 37.
conditions, the inlet total pressure and total temperature were fixed at p01 = 101,325 Pa and T01 = 288.15 K, respectively. Measured and calculated rotor performance maps are presented in Fig. 7(a). The experimental and computed mass-flow rates were normalized using the corresponding
Total pressure ratio
2.20 2.15 2.10 2.05 2.00
Experiment CFD
1.95 1.90 0.90
0.92
0.94 0.96 Normalized mass flow
0.98
1.00
0.98
1.00
Adiabatic efficiency
0.900 0.880 0.860 0.840 0.820
Experiment CFD
0.800 0.780 0.90
0.92
0.94
0.96
Normalized mass flow
Fig. 7(a). Performance maps. Experimental data from AGARD [18].
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Span %
1020 100
100
100
80
80
80
60
60
60
40
40
40
20
20
CFD
20
Exp
0
0 1.7
1.9
2.1
Total pressure ratio
2.3
0 1.1
1.2
1.3
Total temperature ratio
1.4
0.6
0.7
0.8
0.9
1
Adiabatic efficiency
_ m_ choke ¼ 98%. Experimental data from AGARD [18]. Fig. 7(b). Radial plots at station 4: m=
mass-flow at choking condition. The model predicted well the choking mass-flow, giving a value of 20.96 kg/s against the measured one of 20.93 kg/s. The efficiency is slightly underestimated in the peak zone (by about 2%), while the calculated total pressure ratio p04/p01 is almost within the measurement uncertainties in all the operating conditions. Fig. 7(b) shows the spanwise distributions of pitch-averaged total pressure, total temperature and adiabatic efficiency at the outlet for 98% of choking mass flow; calculated profiles are in good agreement with the measurement data. Three-dimensional shock structure and secondary flows (tip-clearance flows, blade wakes, etc.) were also well predicted by the model. Comparisons with experimental data can be found in a recent work of the authors [19]. 4. Results All the modelled rotors were simulated, fixing the same static pressure at the outlet of the computational domain. Even if this is not the appropriate boundary condition for consistently comparing the performance of different designs (each rotor can be at a different aerodynamic-loading level), this approach makes possible a direct comparison with the baseline performance. Fig. 8 compares the predicted efficiency of each new rotor to the computed one of Rotor 37 at the same normalized mass-flow. It is clear that no information can be obtained in this way about the complete performance maps. For each rotor, the mass-flow rate was normalized using the corresponding computed choking mass-flow. For completeness, Fig. 9 shows the predicted choking mass flow rate of different designs. The rotors named as ‘‘others’’ in Figs. 8 and 9 correspond to those modelled rotors which cannot simply be classified as ‘‘forward or backward curved’’, such as the rotor derived from the application of stacking line ‘‘D’’ of Fig. 2. Not very high increments in the overall efficiency were induced by the introduction of a swept stacking line. In Fig. 8(a), the rotors with a leading-edge profile regularly curved downstream (backward swept) are marked with triangles, while the rhombuses indicate all the rotors with a leading edge significantly turned upstream (forward swept). Best
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1021
0.880
Adiabatic efficiency
Rotor 37 0.870
Forward-swept rotors Aft-swept rotors
0.860
Others
0.850
0.840
0.830 0.90
0.92
0.94
0.96
098 .
100 .
0.98
1.00
Normalized mass flow
Swept modelled rotors
Rotor 37
0.870
Forward leaned rotors Backward leaned rotors
0.860
Others
0.850
0.840
0.830 0.90
0.92
0.94
0.96
Normalized mass flow
Leaned modelled rotors Fig. 8. Computed adiabatic efficiency.
21.5 21.3
Backward leaned
21.1
Rotor 37
Others
Aft-swept
21.7
20.9 20.7 20.5 20.3 20.1
Forward swept
Choking flow rate[kg/s]
Adiabatic efficiency
0.880
Forward leaned
19.9
Fig. 9. Choking mass-flow rate of different designs.
1022
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027 MERIDIONAL PLANE
CIRCUMFERENTIAL PLANE
Downstream
100
100
80
80
Span %
Span %
Upstream
60
60
40
40
20
20
0 -0.8
0 -0.4
0
0.4
Axial displacement [cm]
0.8
-3
-1
1
3
Tangential displacement [deg]
Swept rotor
Leaned rotor
Fig. 10. Rotors described.
results were obtained by the backward swept rotors, which achieved an efficiency of about 0.6% higher than that of Rotor 37 referred to the same operating condition. The same rotors have also produced the highest choking mass-flows, with a maximum of about 4% higher compared with the baseline rotor. Similar results were obtained by Abdelhamid et al. [20]. The application of forward sweep, instead, did not provide substantial improvements in the overall efficiency and also resulted in a lower choking mass-flow. As shown in Fig. 8(b), a considerably better performance was achieved using forward leaned rotor-blades, with an efficiency increase of 1.2% above that of Rotor 37. On the other hand, as expected, the backward-leaned curvature was detrimental in most cases. It was also observed that all the leaned rotors produced a very similar choking mass-flow to that of Rotor 37 (Fig. 9). In the following discussion, the backward-swept rotor and the forward-leaned rotor defined in Fig. 10 are analyzed with respect to the original Rotor 37. In Fig. 11, the computed performance maps are compared. Substantial improvements in the adiabatic efficiency were obtained in all operating conditions for both swept and leaned rotors (approximately 1.3% higher than that of Rotor 37 at 99% choking mass flow for the leaned rotor). Moreover, a slightly higher total-pressure ratio was achieved over a large part of the operating range. As shown, the three rotors denote approximately the same stall margin. The near-stall operating point was computed as the last point for which the steady model implemented was able to converge. Fig. 12 compares the computed radial distributions of performance quantities at the peak efficiency condition (close to 99% choking mass flow). At the outer span, Rotor 37 gave a slightly superior efficiency due exclusively to a higher total-pressure ratio delivered. On the other hand, between 30% and 80% span height, for both swept and leaned rotors, a substantially better performance can be observed. This is a direct consequence of the three-dimensional shock structure induced by the blade curvature; in this region which
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1023
2.20
Total pressure ratio
2.15 2.10 2.05 R37 2.00
Leaned rotor Swept rotor
1.95 1.90 0.90
0.92
0.94
0.96
0.98
1.00
Normalized mass flow
Adiabatic efficiency
0.880
0.870
0.860
0.850
0.840
0.830 0.90
0.92
0.94
0.96
0.98
1.00
Normalized mass flow
Fig. 11. Performance maps.
100
100
80
80
80
60
60
60
40
40
20
20
Span %
100
40 R37
20
0 1.85
Sw Ln
1.95 2.05
2.15
Total pressure ratio
2.25
0 1.22
0 1.24
1.26
1.28
1.3
Total temperature ratio
0.7
0.8
0.9
1
Adiabatic efficiency
Fig. 12. Radial plots at outlet – peak efficiency.
differs from the tip region, where there are many flow phenomena which interact with each other leading to a complex flow field, the shock is the flow phenomenon which can mainly influence the local performance of the rotor.
1024
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
Fig. 13 shows the peak-efficiency distribution for the relative Mach number (Mach number distribution relative to the rotor reference frame) inside the blade passage. As apparent, the spatial shape of the shock is clearly influenced by the blade curvature. At the upper half of the span height, the leaned rotor presents a shock front considerably
Fig. 13. Comparison of relative Mach-number distributions – peak efficiency.
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1025
curved downstream in the blade-to-blade view (Fig. 13(b) and (c)); consequently, a slightly aft-swept inclination is also induced in the meridional view (Fig. 13(a)). This certainly helps to reduce aerodynamic shock losses, leading to positive effects on the overall efficiency; on the other hand, the shock interacts with the suction surface more downstream, giving rise to the risk of inducing boundary-layer separation towards the rear. For the swept rotor, the shock presents a more oblique pattern from hub to tip compared with
Fig. 14. Comparison of relative Mach-number distributions – near stall.
1026
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
the unswept rotor, as a consequence of the meridional blade leading-edge curvature; no substantial changes are noticeable in the blade-to-blade views with respect to the Rotor 37. The relative Mach number distribution, at near stall operating condition, is shown in Fig. 14. As expected, all the rotors produce a shock front detached from the blade leading-edge and located further upstream compared with the shock at the corresponding peak-efficiency condition, as consequence of the higher incidence angle due to the lower mass-flow rate. It is interesting to point out that, while the Rotor 37 and the backwardswept rotor gave rise to a similar normal shock situation at both 70% and 90% span, the forward-leaned rotor showed a slightly less detached shock front at 90% span. 5. Conclusions A systematic analysis in order to understand the effects of swept and leaned blades on the aerodynamics of transonic axial-flow compressor rotors was carried out. Many new transonic axially-swept and tangentially-leaned rotors were modelled from the radiallystacked NASA Rotor 37, by changing respectively the meridional and tangential curvatures of the original stacking line. Sweep and lean effects were separately analyzed. An attempt was made to not modify any other design parameter. A commercial CFD code, which solves the Reynolds-averaged Navier–Stokes equations, was successfully validated and used to solve the complex flow-field inside the rotors. The backward swept rotors had a slightly better performance compared with the forward-swept rotors, with a predicted efficiency of about 0.5–0.6% higher than that of Rotor 37 under the same operating-conditions; in addition, a significant higher choking mass-flow was observed. The specific backward-swept rotor analyzed showed the possibility of obtaining a considerable aft-swept shock front with a convenient meridional leadingedge curvature. As expected, a substantial increment in the overall efficiency (up to 1.3% higher than that of Rotor 37) was obtained by applying a stacking line curved toward the direction of rotation (forward lean). A shock front visibly curved downstream on the blade-to-blade plane was observed in all the forward-leaned rotors modelled, with a considerable reduction of shock strength. Very little influence was observed on the choking mass-flow rate. The three-dimensional shock structure represents one of the most important aspects to take into account in order to reduce shock losses and to improve the aerodynamic behaviour of the rotor; it was demonstrated that the use of swept or leaned blades can effectively help to achieve this aim. Further improvements in the overall efficiency can be expected from the simultaneous use of sweep and lean. References [1] Hah C, Puterbaugh SL, Wadia AR. Control of shock structure and secondary-flow field inside transonic compressor rotors through aerodynamic sweep, ASME Paper 98-GT-561, 1998. [2] Wadia AR, Szucs PN, Crall DW. Inner workings of aerodynamic sweep. ASME J Turbomachinery 1998;120(4):671–82. [3] Denton JD, Xu L. The effects of lean and sweep on transonic fan performance, ASME Paper GT-200230327, 2002. [4] Yamaguchi N, Tominaga T, Hattori S, Mitsubishi T. Secondary-loss reduction by forward-skewing of axial compressor rotor blading. In: Proceedings of 1991 Yokohama international gas turbine congress, vol. 2, 1991.
E. Benini, R. Biollo / Applied Energy 84 (2007) 1012–1027
1027
[5] Denton JD, Xu L. The exploitation of three-dimensional flow in turbomachinery design, VKI lecture series 1999–2002 – turbomachinery blade design system, Belgium, 1999. [6] Bergner J, Hennecke DK, Hoeger M, Engel K. Darmstadt Rotor No. 2 – part II design of lean rotor blades, ISROMAC 9, Honolulu, Hawaii, USA, February 10–14, 2002. [7] Ahn CS, Kim KY. Aerodynamic-design optimization of an axial compressor rotor, ASME Paper GT-200230445, 2002. [8] Benini E. Three-dimensional multi-objective design optimization of a transonic compressor rotor. AIAA J Propul Power 2004;20(3):559–65. [9] Reid L, Moore RD. Design and overall performance of four highly-loaded, high-speed inlet stages for an advanced high-pressure-ratio core compressor, NASA TP 1337, 1978. [10] Moore RD, Reid L. Performance of single-stage axial flow transonic compressor with rotor and stator aspect-ratios of 1.19 and 1.26, respectively, and with a design pressure ratio of 2.05, NASA TP 1659, 1980. [11] Wadia AR, Copenhaver WW. An investigation of the effect of cascade area ratios on transonic compressor performance. ASME J Turbomachinery 1996;118(4):760–70. [12] CFX 5.7.1, ANSYS, Inc., Canonsburg, PA 15317, USA. [13] Hutchinson BR, Raithby GD. A multigrid method based on additive correction strategy. Numer HeatTransfer 1986;9:511–37. [14] Launder B, Spalding D. The numerical computation of turbulent flow. Comp Math Appl Mech Eng 1974;3:269–89. [15] CFX TurboGrid 2.2, ANSYS, Inc., Canonsburg, PA 15317, USA. [16] Suder KL, Celestina ML. Experimental and computational investigation of the tip-clearance flow in a transonic axial-compressor rotor. ASME J Turbomachinery 1996;118(2):218–29. [17] Suder KL, Chima RV, Strazisar AJ, Roberts WB. The effect of adding roughness and thickness to a transonic axial-compressor rotor. ASME J Turbomachinery 1995;117(4):491–505. [18] AGARD, CFD validation for propulsion system components, Agard-AR-355, May 1998. [19] Biollo R, Benini E. Validation of a Navier–Stokes solver for CFD computations of transonic compressors, ESDA2006-95318, 2006. [20] Abdelhamid HF, Shreeve RP, Hobson GV. Sweep in a transonic fan-rotor: part 2. CFD and stress analyses. Presented at the 43rd ASME gas-turbine and aeroengine technical congress, exposition and user’s symposium, June 2–5, 1998, Stockholm, Sweden, ASME Paper 98-GU-579.