Non-axisymmetric Endwall Profiling Research Articles

Control mechanism of secondary flow in a turbine cascade with non-axisymmetric endwall profiling under Co-rotating incoming vortex

Abstract Upstream vortex has a significant effect on the secondary flow structure of the downstream turbine in the stage environment. This study investigates the secondary flow structure with non-axisymmetric endwall profiling (NAEW) under the interaction of co-rotating incoming vortex (Vic). A half-delta wing vortex generator is utilized to model Vic. The turbine cascade case which exhibited maximum reduction of the cascade loss with NAEW under no incoming vortex is studied. The mechanism of loss reduction with NAEW under the interaction of Vic is analysed. Vic could decrease the secondary flow near the endwall region by affecting the horseshoe vortex transport in the cascade. However, its loss reduction was lower than the loss increments of Vic itself. The arrival of Vic at the leading edge of the cascade increased the strength of the horseshoe vortex, resulting in a significant increase in loss. Under the interaction of Vic, NAEW decreased the blade loading near endwall region, which resulted in the reduction of cascade loss.

Numerical Investigation of a Turbine Stator with Nonaxisymmetric Endwall Profiling

Numerical Investigation of a Turbine Stator with Nonaxisymmetric Endwall Profiling

Flow physics of highly loaded tandem compressor cascade with non-axisymmetric endwall profiling

Tandem configuration is an effective methodology to reduce flow separation on compressor blade suction surface and to improve blade loading. However, in modern highly loaded cases, corner separation remains as its single blade counterpart. In this study, non-axisymmetric endwall profiling (NAEP) was utilized in a highly loaded tandem cascade (diffusion factor D = 0.69), aiming at reducing its severe corner separation and revealing the unique flow mechanism while NAEP is utilized in tandem cascade. NAEP was designed in both forward (F) blade and rare (R) blade separately, and was investigated numerically in tandem environment. Results show that, NAEP in F blade passage can effectively eliminate the corner separation and reduce loss generation, whereas NAEP in R blade passage has no positive effect on corner separation and even promotes loss production. The optimal NAEP approximately removes the corner separation completely, with loss coefficient reducing by as much as 37.8%. The optimal NAEP for the tandem cascade features optimal axial location at the origin of corner separation. There is an optimal NAEP height (0.02 of blade height), under which NAEP can achieve pretty good control effect while the peak of NAEP varies in a large axial location range. In the tandem configuration, it is found that NAEP transfers blade loading from R blade to F blade; the static pressure increases significantly for the entire cascade, but the static pressure distribution of F blade does not exhibit as the design intent of NAEP. In addition, it is interesting to find that the flow turning near endwall reduces after endwall profiling, which is unique in tandem cascade and is contrast to the view on conventional configuration. On the contrary, NAEP in R blade has no influence on the corner separation of the tandem cascade; due to the decrement of cross-passage pressure gradient for R blade, the flow overturning near endwall reduces.

Effect of Non-Axisymmetric Endwall Profiling on Heat Transfer and Film Cooling Effectiveness of a Transonic Rotor Blade

Abstract Effects of non-axisymmetric endwall profiling on total pressure loss, heat transfer, and film cooling effectiveness of a transonic rotor blade were numerically investigated. The numerical methods, including the turbulence model and grid sensitivity, were validated with the existing experimental data. To reduce the thermal load on endwall, non-axisymmetric endwall profiling near leading edge and at pressure-side corner area was performed with a range of contour amplitudes. Heat transfer and flow fields near the profiled endwalls were analyzed and also compared with the plain endwall configuration. On the profiled endwall, three kinds of cooling holes, i.e., cylindrical holes, rounded-rectangular holes, and elliptical holes, were arranged, and film cooling effect was investigated at three blowing ratios. Results indicate that, with endwall profiling, the area-averaged Stanton number on endwall is reduced by 7.71% and total pressure loss in cascade is reduced by 11.07%. Among three kinds of cooling holes, the arrangement of the elliptical hole performs the best film cooling effect on the profiled endwall. Compared with the plain endwall, non-axisymmetric endwall with elliptical cooling holes improves film cooling coverage by 10.87%, reduces the Stanton number by 8.88%, and increases the net heat flux reduction performance by 4% at M = 0.7.

Control mechanisms of endwall profiling and its comparison with bowed blading on flow field and performance of a highly-loaded compressor cascade

In order to exploit the control mechanism of endwall profiling (EP) on flow field and performance of compressors, this paper carried out investigations on non-axisymmetric endwall profiling (NAEP), axisymmetric endwall profiling (AEP) and the comparison with bowed blading in a highly-loaded compressor cascade. Firstly, the influence mechanism of NAEP on corner separation and its design strategies were investigated. Pattern effect of NAEP, axial effect and height effect of concave/convex of NAEP were studied. Secondly, the influence mechanism of AEP on corner separation and its design strategies were studied. Thirdly, the comparison of flow mechanism between endwall profiling (EP) and bowed blading was carried out. Results show that although all the NAEP patterns can reduce the cross-passage pressure gradient, they exhibit distinct different control effects on corner separation. It is indicated that the radially inward pressure gradient, rather than the cross-passage pressure gradient near the endwall, is the key to control corner separations. The endwall cross-passage pressure gradient remained after AEP. However, the corner separation was also eliminated effectively by AEP with a concave curvature, with loss coefficient reduced by 20.07%. At last, compared with bowed blading, the cascade with EPs exhibits a higher overall performance but a lower control effect on corner separation.

Numerical Investigation and Non-Axisymmetric Endwall Profiling of a Turbine Stage

This paper presents an optimization of a high pressure turbine by constructing non-axisymmetric endwalls to the stator row and the rotor hub. The optimization was quantified by using optimization algorithms based on the multi-objective function. The objective was to increase total-to-total efficiency with the constraint on the mass flow rate equal to the design point value. In order to ensure that global optimum could be achieved, the function of parameters was first approximated through the artificial neural network, and then optimum was achieved by implementing the genetic algorithm. It was adopted through the design and optimization environment of Fine™/Design3D. Three individual treatments of the endwalls were presented. Firstly, the hub and the shroud of the stator were optimized together. Secondly, the hub of the rotor was optimized. Thirdly, the rotor hub was optimized in the presence of the optimized stator. The result of the investigation showed that the optimized shape of the endwalls can significantly help to increase the efficiency up to 0.18% with the help of a reduction of the transverse pressure gradient. The coefficient of secondary kinetic energy, entropy coefficient, spanwise mass averaged entropy were reduced. In order to investigate the periodic effects, the design of the optimized turbine under steady simulations was confirmed through unsteady simulations. The last part of the investigation made sure that the performance improvement remained consistent over the full operating line at off-design conditions by the implementation of non-axisymmetric endwalls.

CFD Investigation on the Application of Optimum Non-Axisymmetric Endwall Profiling for a Vaned Diffuse

In order to improve the performance of a transonic centrifugal compressor stage, non-axisymmetric endwall profiling optimization was conducted for the diffuser under design condition, Artificial Neural Network (ANN) and Genetic Algorithm (GA) were used to execute the optimization with the objective of maximizing the isentropic efficiency of the compressor stage. The influence mechanism of non-axisymmetric endwall profiling on flow field and performance was discussed. Results show non-axisymmetric endwall profiling is an effective way to significantly reduce the flow loss in the diffuser. The total pressure loss of the diffuser decreases by 9.31% and 20.29% for NA0.70 and NA1.40 respectively. The profiled endwall suppresses the flow separation through accelerating the low-energy flow and reducing lateral pressure gradient. The corresponding high vorticity within the flow separation zone is reduced, which delays the formation and development of the flow separation. The diffuser becomes more fore-loaded, the overall blade loading is not affected, and the pressure ratio of the compressor stage is improved as well. At the outlet of the diffuser, the more uniform flow angle and much lower total pressure loss along spanwise are obtained. However, the backflow with high velocity gathering near the shroud of the diffuser makes the mass flow rate decrease and easily induce the stall, which results in the smaller operating range for both profiled endwall.

Non-axisymmetric endwall profiling in a highly loaded compressor cascade

Very few studies of non-axisymmetric endwall profiling in compressor have been conducted in published literatures. This study investigates the control mechanism of non-axisymmetric endwall on the flow field of a highly loaded compressor cascade and the guidelines for the design of non-axisymmetric endwall. First, CFD method was validated with existing experimental data. Then, the design method of non-axisymmetric endwall based on trigonometric function and Bezier curve was introduced. The design strategies of non-axisymmetric endwall considering axial position effect, height effect and different patterns of concave/convex were investigated in detail. Results show that non-axisymmetric endwall is an effective way to control the near endwall flow field of the compressor cascade. With the optimal non-axisymmetric endwall in this study, the loss coefficient of the cascade is reduced by 5.5%, and the cascade exhibited suppressed corner separation, reduced passage vortex and more uniformly distributed outflow angles. The optimal axial position of the concave was at 0.22 axial chord, which was near the onset of corner separation, so the best control effect was obtained. There is an optimal height for the concave of the non-axisymmetric endwall. A lower concave will not have sufficient effect on the cascade, whereas a higher concave will be detrimental on the contrary. It is indicated that the influence mechanism of non-axisymmetric endwall on the compressor cascade is basically derived from the superimposed effect of reduced cross-passage pressure gradient and increased spanwise inward pressure gradient. The cross-passage pressure gradient can be reduced by both concave on the suction side and convex on the pressure side of the endwall. However, concave can introduce spanwise inward pressure gradient, whilst convex can introduce spanwise outward pressure gradient. For better improving the performance of compressor cascades, non-axisymmetric endwall is suggested to be a concave on the entire endwall, which introduces both reduced cross-passage pressure gradient and increased spanwise inward pressure gradient.

An investigation on the effect of pitchwise endwall design in a turbine cascade at different incidence angles

This paper describes the effects of non-axisymmetric endwall profiling on the aerodynamic performance of a linear turbine cascade at different incidence angles. The sinusoidal profiling is carried out with constant profile curvature along the mean streamline path. Three different profiles, with varying hump to dip height, are analyzed numerically and the performances are compared with the planar profile. Reynolds Averaged Navier Stokes (RANS) equations are solved in their conservative form using Finite Volume Method with SST turbulence model. The calculated results indicate that the profiled endwall minimizes the lateral movement of weaker boundary layer fluid from the hub-pressure side corner. In comparison with planar case, the flow deviations are largely contained with endwall profiling but closer to the endwall it enhances the overturning and secondary flow kinetic energy. The reduction in loss coefficient is estimated to be 1.3%, 8.7% and 38% for incidence angles of −10°, nominal and +15° respectively. The sinusoidal profiling has brought down the pitch averaged flow deviation and secondary flow kinetic energy at nominal and positive incidence angles but the impact is insignificant at negative incidence. Profiling minimizes the rolling up of the passage vortex and makes the passage vortex to migrate closer to the endwall. This flow modification brings down the losses in the core flow but enhances the losses near the endwall.

The aerodynamic optimization design of turbine cascade nonaxisymmetric endwall and the midgap influence assessment

The nonaxisymmetric endwall profiling has been proven to be an effective tool to reduce the secondary flow loss in turbomachinery. In the present work, first, without considering the endwall midgap in the real machine, an endwall optimization design procedure for reducing secondary flow losses has been developed, which allowed complete three-dimensional parameterization turbine endwall design. The profile of the endwall has been designed using automatic numerical optimization by means of an improved efficient global optimization algorithm based on kriging surrogate model. Next, a large-scale linear cascade with a low-speed wind tunnel has been chosen for the experimental validation of the optimization results. The experimental measurements and numerical simulations both demonstrated that the total pressure loss and secondary flow intensity were reduced with the nonaxisymmetric endwall used in the cascade passage. Then, in order to evaluate the ability of the optimized nonaxisymmetric endwall with the midgap, the midgap was added in for both the baseline flat endwall and the optimized nonaxisymmetric endwall in the numerical simulations analysis. The entropy generation rates analysis were used for the investigation of loss distribution in the passage. For the cascade in the present work, with the midgap added in, the optimized nonaxisymmetric endwall did not perform as well as the situation without the midgap in the loss reduction. In addition, comparing to the baseline flat endwall, the optimized nonaxisymmetric endwall needed more net leakage flow to avoid the ingress of passage flow into the midgap.

Numerical investigation of inlet boundary layer skew in axial-flow compressor cascade and the corresponding non-axisymmetric end wall profiling

This paper investigates the influence of boundary layer skew on flow structure, total pressure loss, and flow control technique numerically on a high-loaded axial-flow compressor cascade. We have developed two new models respectively about loss evaluation and end wall flow control mechanism for more specific analysis. The result shows that boundary layer skew weakens the secondary flow and delays the generation of passage vortex when incidence approaches 0°. This results in a reduction of total pressure loss mainly (89.4%) due to relieved corner separation. However, as incidence exceeds a certain value (+7°), severe corner separation or even earlier corner stall can be induced by inlet boundary layer skew. Optimization procedure for profiled end wall at inflow condition of +7° incidence is further carried out to investigate the impact of boundary layer skew on flow control technique. The result shows that boundary layer skew should be counted in the optimization design of profiled end wall because of its significant influence on the development of end wall flow. The optimum profiled end walls for cases with and without boundary layer skew show great difference in the manner of end wall flow control. According to the improvement of cascades’ performance, end wall profiling seems more efficient in reducing loss when influenced by the boundary layer skew.

Integrated Combustor Turbine Design for Improved Aerothermal Performance: Effect of Dilution Air Control

The effect of dilution air control in a combustor on the heat load distribution of an axial turbine with nonaxisymmetric endwall profiling is examined. Endwall profiling is a more common design feature in new engine types, due to its effectiveness in reducing secondary flows and their associated losses. In the present work, the effect of dilution air control is examined by using two different circumferentially nonuniform hot-streak shapes; the two cases differ in their spanwise extents either side of the stator and, therefore, represent different approaches for dilution air control. This numerical study details the impact of these two different strategies for dilution air control on the rotor blade heat load distribution. The inlet boundary conditions simulate the experiment that is conducted in the axial research turbine facility LISA at ETH Zurich. A circumferential nonuniformity in the spanwise migration pattern of the hot streak inside the stator is observed that is found to be alleviated by the effect of the endwall profiling. Due to the observed spanwise migration pattern inside the stator the two hot-streak cases result in considerably different heat load distributions on the rotor blade, emphasizing the importance of the integrated combustor turbine approach. Finally, the implications for dilution air control on the liner are discussed for the realization of the simulated hot-streak shapes in real combustors.

Influence of Rim Seal Purge Flow on the Performance of an Endwall-Profiled Axial Turbine

Nonaxisymmetric endwall profiling is a promising method to reduce secondary losses in axial turbines. However, in high-pressure turbines, a small amount of air is ejected at the hub rim seal to prevent the ingestion of hot gases into the cavity between the stator and the rotor disk. This rim seal purge flow has a strong influence on the development of the hub secondary flow structures. This paper presents time-resolved experimental and computational data for a one-and-1/2-stage high work axial turbine, showing the influence of purge flow on the performance of two different nonaxisymmetric endwalls and the axisymmetric baseline case. The experimental total-to-total efficiency assessment reveals that the nonaxisymmetric endwalls lose some of their benefit relative to the baseline case when purge is increased. The first endwall design loses 50% of the efficiency improvement seen with low suction, while the second endwall design exhibits a 34% deterioration. The time-resolved computations show that the rotor dominates the static pressure field at the rim seal exit when purge flow is present. Therefore, the purge flow establishes itself as jets emerging at the blade suction side corner. The jet strength is modulated by the first vane pressure field. The jets introduce circumferential vorticity as they enter the annulus. As the injected fluid is turned around the rotor leading edge, a streamwise vortex component is created. The dominating leakage vortex has the same sense of rotation as the rotor hub passage vortex. The first endwall design causes the strongest circumferential variation in the rim seal exit static pressure field. Therefore, the jets are stronger with this geometry and introduce more vorticity than the other two cases. As a consequence the experimental data at the rotor exit shows the greatest unsteadiness within the rotor hub passage with the first endwall design.

Numerical investigation of a high-subsonic axial-flow compressor rotor with non-axisymmetric hub endwall

The major source of loss in modern compressors is the secondary loss. Non-axisymmetric endwall profile contouring is now a well established design methodology in axial flow turbines. However, flow development in axial compressors is differ from turbines, the effects of non-axisymmetric endwall to axial compressors requires flow analysis in detail. This paper presents both experimental and numerical data to deal with the application of a non-axisymmetric hub endwall in a high-subsonic axial-flow compressor. The aims of the experiment here were to make sure the numerically obtained flow fields is the physical mechanism responsible for the improvement in efficiency, due to the non-axisymmetric hub endwall. The computational results were first compared with available measured data of axisymmetric hub endwall. The results agreed well with the experimental data for estimation of the global performance. The coupled flow of the compressor rotor with non-axisymmetric hub endwall was simulated by a state-of-the-art multi-block flow solver. The non-axisymmetric hub endwall was designed for a subsonic compressor rotor with the help of sine and cosine functions. This type of non-axisymmetric hub endwall was found to have a significant improvement in efficiency of 0.45% approximately and a slightly increase for the total pressure ratio. The fundamental mechanisms of non-axisymmetric hub endwall and their effects on the subsonic axial-flow compressor endwall flow field were analyzed in detail. It is concluded that the non-axisymmetric endwall profiling, though not optimum, can mitigate the secondary flow in the vicinity of the hub endwall, resulting in the improvement of aerodynamic performance of the compressor rotor.

Improving Efficiency of a High Work Turbine Using Nonaxisymmetric Endwalls—Part II: Time-Resolved Flow Physics

This paper is the second part of a two part paper that reports on the improvement of efficiency of a one and a half stage high work axial flow turbine. The first part covered the design of the endwall profiling, as well as a comparison with steady probe data; this part covers the analysis of the time-resolved flow physics. The focus is on the time-resolved flow physics that leads to a total-to-total stage efficiency improvement of 1.0%±0.4%. The investigated geometry is a model of a high work (Δh/U2=2.36), axial shroudless HP turbine. The time-resolved measurements have been acquired upstream and downstream of the rotor using a fast response aerodynamic probe (FRAP). This paper contains a detailed analysis of the secondary flow field that is changed between the axisymmetric and the nonaxisymmetric endwall profiling cases. The flowfield at the exit of the first stator is improved considerably due to the nonaxisymmetric endwall profiling and results in reduced secondary flow and a reduction in loss at both hub and tip, as well as a reduced trailing shed vorticity. The rotor has reduced losses and a reduction in secondary flows mainly at the hub. At the rotor exit, the flow field with nonaxisymmetric endwalls is more homogenous due to the reduction in secondary flows in the two rows upstream of the measurement plane. This confirms that nonaxisymmetric endwall profiling is an effective tool for reducing secondary losses in axial turbines. Using a frozen flow assumption, the time-resolved data are used to estimate the axial velocity gradients, which are then used to evaluate the streamwise vorticity and dissipation. The nonaxisymmetric endwall profiling of the first nozzle guide vane show reductions in dissipation and streamwise vorticity due to the reduced trailing shed vorticity. This smaller vorticity explains the reduction in loss at midspan, which is shown in the first part of the two part paper. This leads to the conclusion that nonaxisymmetric endwall profiling also has the potential of reducing trailing shed vorticity.

Improving Efficiency of a High Work Turbine Using Nonaxisymmetric Endwalls— Part I: Endwall Design and Performance

This paper is the first part of a two part paper reporting the improvement of efficiency of a one-and-half stage high work axial flow turbine by nonaxisymmetric endwall contouring. In this first paper the design of the endwall contours is described, and the computational fluid dynamics (CFD) flow predictions are compared with five-hole-probe measurements. The endwalls have been designed using automatic numerical optimization by means of a sequential quadratic programming algorithm, the flow being computed with the 3D Reynolds averaged Navier-Stokes (RANS) solver TRACE. The aim of the design was to reduce the secondary kinetic energy and secondary losses. The experimental results confirm the improvement of turbine efficiency, showing a stage efficiency benefit of 1%±0.4%, revealing that the improvement is underestimated by CFD. The secondary flow and loss have been significantly reduced in the vane, but improvement of the midspan flow is also observed. Mainly this loss reduction in the first row and the more homogeneous flow is responsible for the overall improvement. Numerical investigations indicate that the transition modeling on the airfoil strongly influences the secondary loss predictions. The results confirm that nonaxisymmetric endwall profiling is an effective method to improve turbine efficiency but that further modeling work is needed to achieve a good predictability.

Optimization of Nonaxisymmetric Endwalls in Compressor S-Shaped Ducts

This paper presents a new design methodology for strutted S-shaped compressor ducts that allows for more aggressive designs while maintaining current levels of duct loss. A baseline duct geometry was selected, which had a radius change to length ratio that is 34% larger than current engine design limits. A large-scale low-speed model of the baseline duct was experimentally tested. The flow in the corner between the hub and the strut was found to separate due to the high local diffusion causing an increase in duct loss. Area ruling was applied to the baseline duct and was predicted to reduce the size and extent of the strut-hub corner separation, but the duct design was compromised. The duct loss coefficient at midpitch was predicted to increase compared with that of the baseline design. Nonaxisymmetric endwall profiling was then used on the duct wall, locally to the strut, to remove the strut-hub corner separation and thus reduce net duct loss, without compromising the duct design away from the strut. The endwall geometry was produced by numerical optimization. It was shown that the net duct loss was insensitive to casing profiling but highly sensitive to hub profiling. The optimal hub geometry was experimentally tested and shown to completely remove endwall strut-hub corner separation. The profiling was found to reduce the net duct loss by 16%. The paper shows that the key benefit to endwall profiling is that it can be used to safely increase the size of the design space in which aeroengine duct designers can operate.

Investigation of a Novel Secondary Flow Feature in a Turbine Cascade With End Wall Profiling

A novel secondary flow feature, previously unreported for turbine blading as far as the authors are aware, has been discovered. It has been found that it is possible to separate part of the inlet boundary layer on the blade row end wall as it is being over-turned and rolled up into the passage vortex. This flow feature has been discovered during a continuing investigation into the aerodynamic effects of non-axisymmetric end wall profiling. Previous work, using the low speed linear cascade at Durham University, has shown the potential of end wall profiling for reducing secondary losses. The latest study, the results of which are described here, was undertaken to determine the limits of what end wall profiling can achieve. The flow has been investigated in detail with pressure probe traversing and surface flow visualization. This has found that the inlet boundary locally separates, on the early suction side of the passage, generating significant extra loss which feeds directly into the core of the passage vortex. The presence of this new feature gives rise to the unexpected result that the secondary flow, as determined by the exit flow angle deviations and levels of secondary kinetic energy, can be reduced while at the same time the loss is increased. CFD was found to calculate the secondary flows moderately well compared with measurements. However, CFD did not predict this new feature, nor the increase in loss it caused. It is concluded that the application of non-axisymmetric end wall profiling, although it has been shown to be highly beneficial, can give rise to adverse features that current CFD tools are unable to predict. Improvements to CFD capability are required in order to be able to avoid such features, and obtain the full potential of end wall profiling.

Improving the Efficiency of the Trent 500-HP Turbine Using Nonaxisymmetric End Walls—Part I: Turbine Design

This paper describes the redesign of the HP turbine of the Rolls-Royce Trent 500 engine, making use of nonaxisymmetric end walls. The original, datum turbine used conventional axisymmetric end walls, while the vane and (shrouded) rotor aerofoil profiles were nominally the same for the two designs. Previous research on the large-scale, low-speed linear cascade at Durham University (see Hartland et al., 1998, “Non-Axisymmetric End Wall Profiling in a Turbine Cascade,” ASME 98–GT-525), had already demonstrated significant potential for reducing turbine secondary losses using nonaxisymmetric end walls-by about one third. This paper shows how a methodology was derived from the results of this research and applied to the design of the single-stage Trent 500-HP turbine (model rig). In particular, the application of a new linear design system for the parametric definition of these end wall shapes (described in Harvey et al., 1999, “Non-Axisymmetric Turbine End Wall Design: Part I Three-Dimensional Linear Design System,” ASME 99–GT-337) is discussed in detail.

Nonaxisymmetric Turbine End Wall Design: Part I— Three-Dimensional Linear Design System

A linear design system, already in use for the forward and inverse design of three-dimensional turbine aerofoils, has been extended for the design of their end walls. This paper shows how this method has been applied to the design of a nonaxisymmetric end wall for a turbine rotor blade in linear cascade. The calculations show that nonaxisymmetric end wall profiling is a powerful tool for reducing secondary flows, in particular the secondary kinetic energy and exit angle deviations. Simple end wall profiling is shown to be at least as beneficial aerodynamically as the now standard techniques of differentially skewing aerofoil sections up the span, and (compound) leaning of the aerofoil. A design is presented that combines a number of end wall features aimed at reducing secondary loss and flow deviation. The experimental study of this geometry, aimed at validating the design method, is the subject of the second part of this paper. The effects of end wall perturbations on the flow field are calculated using a three-dimensional pressure correction based Reynolds-averaged Navier–Stokes CFD code. These calculations are normally performed overnight on a cluster of work stations. The design system then calculates the relationships between perturbations in the end wall and resulting changes in the flow field. With these available, linear superposition theory is used to enable the designer to investigate quickly the effect on the flow field of many combinations of end wall shapes (a matter of minutes for each shape). [S0889-504X(00)00902-8]