Influence of fillet shapes on secondary flow field in a transonic axial flow turbine stage

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Influence of fillet shapes on secondary flow field in a transonic axial flow turbine stage K. Ananthakrishnan ∗ , M. Govardhan TurboMachines Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-36, India

a r t i c l e

i n f o

Article history: Received 11 April 2018 Received in revised form 25 June 2018 Accepted 28 August 2018 Available online xxxx Keywords: Transonic turbine stage Secondary flow Leading edge fillet

a b s t r a c t Numerical experiments were conducted to investigate the effect of leading edge modifications via fillet shapes near vane/blade-endwall juncture in a transonic environment within the highly loaded high pressure turbine stage. The investigated fillet shapes were designed based on geometric parameters: leading edge radius and included angle. The geometrical modifications were achieved to achieve variation in fillet radii at vane/blade endwall juncture along the stream-wise direction, namely variable fillet and constant fillet. Further their influences were studied in both nozzle guide vane and rotor passage secondary flow field. Computational Fluid Dynamics (CFD) method was used to resolve the flow features inside the turbine passage for planar and fillet cases. The presented data highlight the secondary flow features and their behavior using topological properties of flow field aided with the streamline and isocontour plots. The flow-field results show a significant reduction in the total pressure losses associated with the horse shoe vortex near the leading edge region as the fillet radii are varied. Overall in both vane and rotor passages, variable fillet outperforms the constant fillet by reducing the penetration length of three-dimensional regimes along its span, mitigating the boundary layer growth and improving the loss coefficients. © 2018 Elsevier Masson SAS. All rights reserved.

1. Introduction One of the primary goals that drive the design outline of a highly effective modern gas turbine engines is to have less secondary flow losses in their flow passages, irrespective of the stage. They comprise horse-shoe vortex, passage vortex, tip leakage vortex and corner vortices, which are for the most part considered as detrimental factors to the reliable operation and performance of gas turbine engines. They are responsible for considerable (30–50%) total pressure losses [1], especially in the high pressure stage, i.e., low aspect ratio blades. Thus, studies related to secondary flow losses and its reduction in turbine passages received a constant source of attention from research community for the last forty years. A better understanding of secondary flow phenomenon happening near the endwalls was obtained with the help of several secondary flow physical models, e.g., Wang et al. [2], and from some excellent reviews and documents by Sieverding [3], Langston [4] and many others. See Fig. 1. Further, availability of present day high-fidelity eddy resolving simulations helps in capturing complex flow field near the endwalls [5]. From previous

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Corresponding author. E-mail addresses: [email protected] (K. Ananthakrishnan), [email protected] (M. Govardhan). https://doi.org/10.1016/j.ast.2018.08.040 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

state of art works, it is evident that the formation of horseshoe vortex and cross flow across the passage due to traverse pressure gradient has to be eliminated or reduced to have low losses associated with these three-dimensional flow structures. Based on whether the energy is needed or not, to operate a flow control device, loss reduction techniques are categorized into active and passive ones. Active control achieves this reduction by expenditure of external energy [6,7]. While, passive denotes flow control that involves making changes in geometry, whereas the active means one that requires external energy requirement. Several novel passive techniques have been reported in literature for reducing the secondary flow losses like leading edge fillet [8–13], endwall fence [14], three-dimensional blade having lean, sweep and bowing [15,16], axisymmetric, non-axisymmetric endwall contouring [17,18] and recently bio-inspired micro-texture on rotor blade [19]. Passive control method gains more attention than the former for its simplicity and absence of additional energy requirement. It was observed that application of leading edge modification at the vane/blade-endwall juncture using fillet shapes shows potential in reducing horse shoe vortex formation [8–12]. For designing the leading edge fillet several ingenious methodologies were put into practice. In the mid 1980s, reduction in interference drag at the wing–fuselage juncture was highlighted by using leading edge modifications in terms of fillet shapes by Kubendran and

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Fig. 1. Secondary flow schematic model of a turbine cascade from Wang et al. [2].

Harvey [8]. Using flow measurements at the wake region, they reported the evidence of reduction in drag as well as improvement in flow characteristics downstream of the juncture when fillet was applied. Later, Davenport et al. [9] came up with the design of leading edge fillet in terms of large fairing in the corner between the wing nose and the body surface. They claimed that at zero angle of attack, it eliminates leading edge separation. Benner et al. [10] re-examined the influence of leading edge geometry on profile loss. An improvement in the co-relation for off-design profile loss is proposed as the function of leading edge radius and wedge angle as the parameter. Sauer et al. [11] proposed the design of leading edge bulb to weaken the passage vortex by intensifying the suction side leg of a horseshoe vortex. Zess and Thole [12] put forward a fillet design for nozzle guide vane having asymmetric variation on both pressure and suction sides having dimensions of 1δ high and 2δ long. The fillet design effectively reduced the effect of total pressure gradient by accelerating the incoming boundary layer fluid. Shih and Lin [20] carried out an investigation on fillet modifications on the basis of how the fillet thickness fades over the airfoil and the endwall. Mahmood et al. [21] investigated experimentally the fillet designs having linear and parabolic profile variations. Although reduction in total pressure loss coefficient and vorticity was reported, linear fillet profile shows more prominent reductions in pressure loss coefficients and pitch angles. Eric Lyall et al. [22] indicated that endwall losses can be reduced by reducing the stagger angle. Such a design modification resulted in a fillet shape that results in 23% reduction in endwall losses. Recently Wei et al. [13] presented a design of leading edge fillet based on teardrop curves. From their experimental and numerical investigations, it was indicated that the present-day numerical capabilities offer a reliable prediction of endwall secondary flows. Despite the fact that various leading edge modifications and their effect on endwall losses were presented in the past, their analysis were confined to subsonic regime predominantly on linear cascade environment [8,21] having typical profiles of highly loaded low pressure turbine [22,13]. But this view of secondary flows are overstated compared to realistic engine contraction [23]. Turgut et al. [24,18] studied the effect of leading edge modifications along with non-axisymmetric endwall contouring in low

speed turbine stage facility. Using both experimental and computational investigations, they reported that LE fillet and contoured endwall effectively reduced the horseshoe vortex and secondary flows respectively in the NGV passage. Few studies were carried out in transonic regime; for example, Saha et al. [25] presented the effect of leading edge fillet and the influence of pre-history on aerodynamic losses of a transonic NGV in cascade environment. Further information regarding the leading edge modifications and its effect on high pressure turbine stage environment operating in transonic regime is sparse. Therefore, an effort is made by the present authors to study the influence of leading edge modification in the NGV of high pressure transonic turbine stage. Fillet designs having constant and variable radius at the vane/blade endwall juncture was investigated. It was shown that fillets having variable radius outperforms the constant on [26]. In continuation of the investigation, the present work attempts to gain further insight into the way in which fillet shapes affects the secondary flow losses of HP turbine stage. Further the effects of fillet modifications on both NGV and blade endwall junctures is investigated. 2. Numerical approach and its verification The present steady state numerical investigations are intended to describe the secondary flow mechanisms occurring inside the transonic high pressure turbine stage and the effects of different fillet shapes on it. Geometric parameters of baseline model, HP turbine stage is shown in Table 1. The computational domain of HP turbine stage as shown in Fig. 2 is modeled using Autogrid5, Numeca 11.1. Three dimensional viscous compressible flow calculations were solved using ANSYS CFX 14.5 CFD code [27,28]. It solves the compressible Navier Stokes equation in the finite volume formulation using node centered approach. Real gas properties are used to take into account specific heat variation with respect to temperature. Total energy heat transfer model was used to predict the temperature field. The effect of turbulence on the flow field of HP turbine was resolved using SST k–ω , two equation eddy viscosity turbulence closure model. It is worthwhile to mention that from some previous studies [14,18,29,28], it was found that SST Model is the

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Table 1 Specification of high pressure turbine stage. Parameters

Nozzle guide vane

Rotor blade

Axial chord, C ax , mm Aspect ratio, h/C ax Turning angle, deg R hub / R tip Pitch/C ax Throat/Pitch Tip clearance Blade count Rotational speed Mass flow rate Inlet total pressure Inlet total temperature

45.87 0.545 71 0.917 1.168 0.326 – 34 –

33.84 1.019 110.5 1.126 0.738 0.4847 0.56 mm 73 15976 rpm 61 kg/s 23.59 bar 1865 K

Fig. 3. Grid independence study carried for HP turbine stage.

Fig. 2. Computational domain of HP turbine stage.

most suitable model for analyzing the secondary flows. Profiles of total pressure and temperature with the turbulent intensity of 8% were specified at inlet plane, placed at 1.5 chords upstream of the vane leading edge. The outlet plane was placed at two chords downstream of rotor trailing edge and mass flow rate was set as the boundary condition. The no-slip boundary conditions was employed on all the walls of the domain treated as adiabatic while solving the energy equation. Rotor shroud wall was mentioned as the counter rotating wall to make shroud as stationary in stationary frame of reference. Periodic interface “rotational periodicity” was used to truncate multiple identical passages into a single passage to reduce the computational cost. Multiple passages in a row are required to keep the ensemble pitch ratio close to unity, as the number of rotor blades falls in a prime number count. The present CFD solver does the flux and profile scaling automatically based on the pitch ratio. By modeling one vane to two rotor passages configuration, pitch ratio of 0.932 close to unity was maintained. Connections between stationary NGV and rotor components was made using the frame change interface model “Frozen rotor”. Steady state simulations were computed until the convergence limit of 10−5 was attained in residual values associated with all the governing equations. Typically convergence is achieved between 100 to 150 iterations. Detailed sensitivity evaluation over spatial discretization error and turbulent models was carried out for all computational mesh involved in the investigation. To keep it brief, grid sensitivity analysis involved with the HP turbine stage is shown in Fig. 3. Five sets of computational grids were generated. Node count along cir-

cumferential and axial direction are varied by the ratio of two, compared to the previous coarse mesh. On spanwise direction, computational domain discretized with 107 nodes and it kept constant. 3-D governing equations were solved for all the above mentioned grids. Variation of η , shaft power, total temperature and pressure across the turbine stage was evaluated with respect to node counts. As the grid resolution increases, variation between these parameters became insignificant. Grid independent solutions attained for grid size of 6 million (≈ 2 million) for each passage. Open test cases (VKI-LS89 [30] and E-TU/4 test case [31]) experimental results were used for verifying and validating the application procedure of ANSYS CFX 14.5® CFD code. VKI-LS89 configuration is a 2-D profile, typical representative of modern aero engine’s NGV of HP turbine. Two test conditions MUR43 and MUR47 corresponding to exit Mach number of 0.84 and 1.02 respectively was used. A detailed description of the test cases was given in VKI-Technote 175 [30]. A fair agreement achieved between experimental and numerical results is illustrated using Figs. 4a and 4b. Fig. 4a shows the qualitative comparison between normal shock using schlieren and numerical computation with the help of density gradient contours. Further, pressure loading around the blade at the operating conditions near high subsonic and transonic case is shown in Fig. 4b. Further E-TU/4 4-stage turbine [31] was used for verifying stage calculation because of its lower flow coefficient resulting in strong three-dimensionality flow nature. The blading was of the free vortex type subjected to high twist along the span, the nominal mass flow rate was 7.8 kg/s and the nominal rotational speed was 7500 rpm. Design point was selected at n/n0 and m/m0 at 0.96 and 0.87 as the operating condition corresponding to 6.786 kg/s and 7200 rpm respectively. Numerical prediction of exit angle distribution and the total pressure loss coefficient at stator and the rotor outlet was compared with the experimental results (Fig. 5). Good agreement with experimental results was obtained at stator exit field, whereas at the rotor exit, deviation near the hub region was observed. This is due to the geometrical uncertainty comes in the form of lofting the 2-dimensional profile along its initial 10% of span from the hub. Since the profile coordinates near the hub is not available i.e. (2-d profile coordinates available near the hub at 140 mm while hub radius is 135 mm) it is extrapolated to intersect with hub radius. Near the shroud side, fair agreement with the experimental results is obtained. Further, to improve the confidence in the numerical predictions, the numerical model was solved us-

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Fig. 5. Comparison of exit angle and total pressure loss coefficient distribution along the span at stator and rotor exit.

Table 2 Comparison of design parameters of HP turbine stage.

Fig. 4. (a) Shock wave structure comparison using schileren visualization at M 2is = 1.03 and density gradient contours at M 2is = 1.02, (b) blade loading comparison for test cases MUR43 & MUR47.

Parameters

ANSYS CFX

FINETurbo

Flow function Specific output function Loading factor Flow coefficient Pressure ratio Isentropic efficiency

1.1332 237.20 1.879 0.548 2.7887 90.594

1.1312 238.066 1.899 0.55 2.781 90.203

ing another CFD solver, Numeca FineTurbo 11.1. Table 2 shows the global parameters associated with the HP turbine stage along with the predictions of different numerical solvers. 3. Vane/blade endwall juncture fillet design The fillet generation process was carried out based on design parameters leading edge radius (R) and included angle (θ ) as described in Fig. 6. Included angle is the angle form between the endwall and the tangent to the leading edge circle. For the given fillet radius value, leading edge circle tangent with the blade surface and the end walls was drawn. It determines the location on the vane/blade span where the fillet starts. Based on including angle (θ ), the intersecting point between the tangent line and the arc was calculated. 2-dimensional profile of was generated by extending the tangent line from this point to the hub. Finally, three dimensional shape of the fillet was generated by sweeping the two-dimensional profile around the whole vane/blade endwall juncture. The geometrical modifications investigated in the present numerical study are to understand how the variation in fillet radii at vane/blade endwall juncture along the stream-wise direction af-

Fig. 6. Design parameters for fillet design.

fects the secondary flow field. Simulated fillet configurations were categorized into three cases: Case I: No fillet (baseline), Case II: Constant fillet where the radii of the fillet was uniform throughout vane/blade endwall juncture from the leading edge to trailing edge and Case III: Variable fillet where the radii of the fillet was varied from leading edge to trailing edge. Variable fillet differs from constant fillet cases in terms of exclusion of fillet near trailing edge

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Table 3 Details of fillet cases investigated. Category

Case I

Case II

Fillet type

No fillet

Constant fillet

Case III

Cases

Base case

1-20-LTE; 2-20-LTE; 3-20-LTE; 4-20-LTE; 5-20-LTE;

Variable fillet

1-30-LTE; 2-30-LTE; 3-30-LTE; 4-30-LTE; 5-30-LTE;

1-45-LTE 2-45-LTE 3-45-LTE 4-45-LTE 5-45-LTE

1-20-LE; 2-20-LE; 3-20-LE; 4-20-LE; 5-20-LE;

1-30-LE; 2-30-LE; 3-30-LE; 4-30-LE; 5-30-LE;

1-45-LE 2-45-LE 3-45-LE 4-45-LE 5-45-LE

Note: 0, 1, 2, 3, 4, 5 denotes leading edge radius (R); 20, 30, 45 deg denotes including angle (θ ); LTE indicates leading & trailing edge; LE indicates leading edge only.

Fig. 7. Geometrical models of NGV and rotor blade with different fillet shapes.

region. To resemble the real world situation, 0.5 mm fillet was modeled at the trailing edge and was kept constant for all cases in variable fillet cases. For Case III, transition between the leading edge radius from leading edge region to trailing edge region was achieved in a linear fashion. Further, the impact of the including angle and leading edge radius variation on Case II and III were also investigated. Limitations involved with the CFD griding and manufacturing feasibility were taken into care while deciding the range of design parameters of fillet design. Leading edge radius (R) was varied from 0 to 5 mm and the including angle (θ ) was varied from 20 to 45 deg. Combinations of these design parameters are exploited in this study. Baseline case is represented by the leading edge radius of 0 mm and 90 deg included angle. Table 3 shows the numerical values of design parameters exploited in the current investigation to understand the effectiveness of fillet application over secondary flow losses. Geometrical variation produced by these design variables over endwall juncture is visually shown in Fig. 7. While designing fillet on the endwall juncture near the nozzle guide vane and rotor blade, sensitivity analysis on a number of layers to be placed on fillet region was undertaken. Region occupied by fillet along the span direction increases as the leading edge radius is increased and does not depend on including angle. It is observed that number of layers placed in the fillet region

along its span has an influential effect on loss predictions. For each leading edge radius, sensitivity analysis on total pressure loss coefficient over five sets of layer variations (21, 25, 29, 33, 37) has been carried out. For 3 and 5 mm radius fillet, the variation in loss coefficient becomes insignificant after 29 layers and 33 layers respectively with the difference falling in the fourth significant digit (0.000164 and 0.000168 respectively). Naming format used in the current study is as follows: 3-30-LE: LE radius ( R = 3 mm) – including angle (θ = 30 deg) – applied only at LE and 3-30-LTE: LE radius (3 mm) – including angle (θ = 30 deg) – applied at both LE and TE. 4. Results and discussion 4.1. Flow field of high pressure turbine stage From Midspan flow field of HP turbine stage, a clear picture of transonic flow phenomenons; strong expansion across the stage, compression shock wave, boundary layer growth and the flow mixing in the wake region can be derived. From contours of Mach number (Fig. 8) strong acceleration of the flow field across the NGV and rotor passage is seen. Sonic line (yellow color) which acts as the boundary between subsonic and supersonic region is shown. Its curvy shape is due to the fact that delay in achieving the pressure relate to sonic condition on pressure surface. Since

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Fig. 8. Contours of Mach number at midspan of HP turbine stage. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

the pressure was only related to total pressure in isentropic region away from the surfaces. Accelerating flow field is terminated by the normal shock wave and its presence is confirmed from the reduction of Mach number to subsonic values. It impinges on the suction surface of the adjacent blade. While the suction side shock emanating from trailing edge is chopped by the rotor blade. Generally, the vortical structures associated with the highspeed blade passages have resemblance similar to the subsonic blade passages. The secondary flow field strongly depends on Mach number variation as its increment reduces the impact of secondary flow losses over total stage losses. This distinctiveness comes in terms of reduction in secondary flow penetration depth, exit angle deviations as well as the strength and size of passage vortex. 4.2. Nozzle guide vane passage In this present study, the topological theory is used to understand the secondary flow features associated with the turbine passages. Skin friction line patterns enable this better interpretation as it describes the flow field immediately on the NGV surface (Fig. 9).

Wall shear values are imposed on the surface in aiding the understanding. Singular point is one at which vector magnitude reaches zero and its direction becomes indeterminate. They are classified into saddle point (S) and nodes according to the field lines behavior near its vicinity. Nodes are further classified into two types: nodal points (N) and foci. A finite number of singular points occur in the vector field and some are highlighted in the upcoming discussion. Lighthill [32] proposed the global view of three dimensional separation using the convergence of limiting streamlines as the necessary criterion. The incoming flow belongs to boundary layer gets separated in front of the NGV along the separation line (S1). Emergence of the line of separation from saddle point (S) provides the existence of flow separation as shown in Fig. 9a. Skin friction lines lies on either side tend to turn abruptly and converge on the separation line. An nodal point of attachment (N) is exist in front of leading edge which feeds the flow downstream of separation line (S1). The separation surface from (S1) rolls up the incoming boundary layer into two legs of horse-shoe vortex surrounding the leading edge. The suction side leg of HS vortex (S1s) as it swept towards the suction side due to traverse pressure gradient as it flows downstream. While the pressure side leg (S1p) driven across the passage to reach the adjacent blade suction surface. These two legs grow in its shape and strength after interacting with one another as if flows downstream towards the passage exit results in passage vortex. The separation line associated with this growth on suction surface is indicated in the topological interpretation as S2h and S2t (Fig. 9b). The region enclosed within S2h and S2t and endwalls are subjected to three dimensional flow structures. Using low wall shear values, foot traces of the trailing edge shock wave and their reflection over the adjacent vane suction surface are clearly revealed. Due to the strong expansion, fully developed turbulent boundary layer resists the formation of separation bubble in the midspan region. This improves the two dimensionality of the flow throughout the span except close to the endwalls. However, near trailing edge, shock wave is strong enough to create a separation bubble. 4.2.1. Rotor blade passage When endwall boundary layer approaches the leading edge of rotor blade, it undergoes a three dimensional separation from the endwall as it cannot negotiate the adverse pressure gradient. But it differs in terms of higher magnitude compared to NGV vortical system due to the presence of unsteadiness in the flow field

Fig. 9. Limiting streamlines drawn near NGV (a) leading edge, (b) suction surface.

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Fig. 10. Limiting streamlines drawn near rotor blade (a) leading edge, (b) suction surface.

Fig. 11. Horse shoe vortex formation and rotor total pressure loss coefficient at different stream wise planes.

due to vane rotor interactions. Above mentioned phenomenon is highlighted using topological property near the leading edge and suction surface (Fig. 10). The separation process associated with the saddle point (S) (Fig. 10a) creates the reverse flow in front of the leading edge, bringing the higher momentum fluid outside the boundary layer close to the endwall. As this reverse flow convects around the leading edge, its interaction with the streamwise flow results in the formation of vortex system with two legs resembling the shape horse-shoe. The separation (detachment) lines associated with the suction leg (S1(SL)) and pressure leg (S1(PL)) along with separation line of attachment near the leading edge are shown in Fig. 10a. Another saddle point (S) exist along with its separation lines S4 (sl and pl) above the attachment node (Na) which feeds the flow downstream of S1. From limiting streamlines pattern on suction side (Fig. 10b), movement of separation lines S1, S3, S4 (near hub) and S5, S7, S8 (near shroud) towards the midspan region is seen. Footprints of trailing edge shock from adjacent blade is shown using low wall shear values. Shock boundary layer interactions initiate the week separations (S5 & S7) and its interactions not extended till the endwalls. Spanwise extent of secondary flow (penetration length) is greater i.e. around 50% of span when compared to vane. Further it gets complicated due to interactions of

tip leakage flow and shock boundary layer interactions. This vortex system is responsible for the total pressure reduction, hence efficiency, as it dissipates the useful energy from the fluid which otherwise can be used to produce useful torque. A flow feature that reduces the efficiency of the component is called as loss and the common way to evaluate is total pressure loss coefficient. For rotating component it is based on relative total pressure. Since profile loss remains same for all cases, reduction in the horse shoe vortex strength leads to the improvement in rotor performance.

R LC =

P 02rel − P 03rel P 02rel − P 3

(1)

Overall, the secondary flow field associated with the rotor passage is shown in Fig. 11 using streamlines. Fig. 11a shows the horseshoe vortex along with its two legs near the hub side endwall. Concentration of high loss region associated with higher turbulent kinetic energy value seems to be confined to the vortex core region. Along the stream-wise direction, the migration of pressure side leg vortex from the pressure side to the suction side and its transformation into the passage vortex is visualized. The pressureside leg of the horseshoe vortex system is a dominant constituent of the passage vortex structure. Planes at different stream-wise distances (6%, 10%, 14% and 18%) of rotor passage are used for this

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Fig. 12. Limiting streamline pattern near the suction surface of NGV for the base case.

purpose. Increment in the size of the vortex core while it is migrating from the pressure side to suction side of the blade passage due to transverse pressure gradient was observed. Relative total pressure loss coefficient values are plotted in various stream-wise planes, and higher values are confined to the vortex core region. Near the tip side (Fig. 11b), the vortex structure pattern gets more complicated due to the addition of a tip clearance vortex. Once the pressure side leg gets driven across the passage, it was observed to collide and combine with or wrap around the suction side leg of the horse shoe vortex, and together they climb the suction surface toward the midspan. It results in the growth of its size. Corner vortex is seen to originate near the meeting place of two legs of HS vortex and tip clearance vortex driven by the pressure difference between pressure side and suction side seems to roll over all the vortical structures. 4.3. Effect of fillet shapes on NGV endwall flows Discriminating features of the limiting streamline pattern drawn over the suction side of the nozzle guide vane for base line case and various fillet cases are discussed in this section. Baseline case (Fig. 12a) shows an interesting phenomenon of upward and downward flow near the hub and shroud endwalls respectively due to radial pressure gradient. The region enclosed between the separation line of detachment and the endwalls is subjected to the three dimensional flows. Secondary flow strength is measured in terms of height of these three dimensional regions along the span. Further, the effect of shock structures is seen by the strong deflection of the skin friction lines near the shock foot regions. They turn across the shock wave and then continue downstream in the outlet flow direction. The lines crossing the shock traces indicate shock boundary layer interactions without separation. Penetration length (Fig. 12a) is defined as the region enclosed between the separation lines formed on the suction surface and the endwall corner. This area encompasses three-dimensional vortical structures and its influence on exit flow field of NGV is shown in Fig. 12b. By suppressing the penetration length, spanwise extent of the vortical structures can be reduced, leading to reduced losses near the hub and shroud walls. Fig. 13 represents the plot of penetration length occurred in terms of percentage of the span for different fillet cases having variations in leading edge radius and included angle. Penetration length occupies more percentage of span near the shroud region when compared to the hub region irrespective of constant and variable radii fillet cases. This clearly illustrates that the secondary flow field is more prominent in spatial dispersion near the shroud side when compared to the hub. As the leading edge radius is increased for the fillet shapes, there is an increment in the penetration length, i.e., the region affected by the passage vortical region

Fig. 13. Effect of leading edge radius and minimum angle over penetration length for constant and variable radii fillet cases.

near the hub and shroud endwalls. This increment is more prominent near the hub for both fillet variations (Case II and Case III) as these shape modifications provide acceleration to the fluid near the endwall juncture. This helps the fluid inside the boundary layer to gain its momentum, which results in overcoming the flow separation near the trailing edge due to shock interactions. The above mentioned phenomenon is shown in Fig. 14 based on the pressure contours over the endwalls of NGV. Locations of shock wave can be visualized from the concentrated iso-pressure lines formed on the suction surface. For baseline case, the shock wave near suction side trailing edge is strong enough to separate the boundary layer fluid away from the surface. But the application of fillet shape near the NGV/endwall juncture helps in overcoming the occurrence of separation bubble. Variation in included angle (20 to 45 deg) is shown for 3 mm leading edge radius (Fig. 13) to understand its influence over the height of the three-dimensional region. It was observed that the included angle has an insignificant contribution to it in both fillet cases (Case II and Case III). Effectiveness of variable fillet (Case III) over constant fillet (Case II) in improving the losses is shown by comparing the to-

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Fig. 14. Static pressure contours over the surfaces of hub and NGV (a) baseline, (b) endwalls and NGV.

4.4. Effect of fillet shapes on rotor endwall flows

Fig. 15. Comparison of total pressure loss coefficient between constant and variable radii fillet with base case having no fillet application.

tal pressure loss coefficient of the NGV passage. Total pressure loss coefficient is defined as the ratio of loss in total pressure across the NGV normalized with the maximum dynamic head available.

T P LC =

P 01 − P 02 P 01 − P 2

(2)

It is clearly evident from Fig. 15 that the constant fillet cases deteriorate the performance as the leading edge radius is increased. As the fillet radius at the leading edge of NGV increased for Case II, skewing of the boundary layer got pronounced thereby displacing the separated flow structures in a manner that the effective shape of the NGV at the endwalls got increased as previously highlighted by Devenport et al. [9]. This causes the penetration length to reach higher values as discussed in Fig. 13. Variable fillet (Case III) outperforms the constant fillet case (Case II) as well as the base case (Case I) at all fillet radii and the maximum reduction is seen at higher radius of 5 mm as compared to Case III. This improvement is attributed to the reduction in the fillet radius in the stream-wise direction just downstream of the leading edge. Further, the effect of the included angle shows an insignificant contribution to the loss coefficients on both fillet cases (II and III). When compared to Case I, maximum improvement is achieved for the lower included angle and radius. From this investigation, it can be concluded that the application of fillet shapes is mandatory in NGV/endwall juncture with the lowest feasible radii taking in account the manufacturing constraint.

Further numerical investigations were carried out in the rotor domain to understand the effect of employing fillet shapes fully around the rotor blade (Case II: constant fillet) and only near the leading edge (Case III: variable fillet). Further their effects on included angle (20 to 45 deg) over secondary flow losses were also studied. For variable radii fillet cases, 0.5 mm fillet was applied at the trailing edge to resemble the real blade endwall juncture. Computational difficulty drives the selection of included angles with a minimum value of 20 deg as mesh skewness level degraded abruptly. Maximum leading edge radius is limited to 5 mm due to the constraints in rotor-vane gap as well as the area blockage. Vector plot (Fig. 16) shows the horse shoe vortex formation near the leading edge and its vortical core region has higher concentration of turbulent kinetic energy (TKE). It indicates the kinetic energy associated with the vortical motions. Pressure and suction side legs of horse shoe vortex are shown in Figs. 16b–16d using streamlines injected from the upstream. Streamlines are superimposed with the contour value of TKE to visualize the swirling strength of the vortices while migrating across the passage. Due to the application of fillet shapes, the saddle point at which the flow separates near the leading edge region moves closer to the blade. Further it was observed that as the included angle decreases from 45 to 20 deg, the swirling strength of pressure-side leg vortex reduces and the least is observed for 20 deg case (Fig. 16d). So in the following investigations, results of fillet shapes having included angle of 20 deg are highlighted. The boundary layer development at the exit of the hub for constant radius fillet case was larger when compared to the variable fillet base case as well as the base case. For the variable fillet case and base case, from schematic view Fig. 17a the passage area increases locally from the throat region to the trailing edge of the pressure surface, which leads to a sudden deceleration of flow. As a consequence, the boundary layer on the suction surface and the endwall grows downstream of the throat. Due to constant fillet application, i.e., Case II, the increment in the passage area (from the throat to the trailing edge) is more when compared to the base case. As a result, boundary layer becomes thicker downstream of the throat in the stream-wise direction and this is shown in Fig. 17 plotted at 1.15C ax streamwise location. Consequently, the total pressure loss associated with this growth at the endwall for the constant radii fillet case is larger compared to the variable radii fillet case. Similar trends in increment in loss coefficients were observed near the shroud side also. The effect of fillet radii variation near the leading edge region of rotor blade was investigated for 0 to 5 mm radius at constant included angle 20 deg. As the leading edge radius determines the height and length of fillet, its maximum value was limited to 5 mm based on the constraint over the axial gap between NGV and rotor

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Fig. 16. (a) Contours of TKE near the leading edge plane of rotor blade and Legs of horseshoe vortex mapped with turbulent kinetic energy values near, (b) Case I base case, (c) 3-45-LTE, (d) 3-20-LTE.

Fig. 17. Boundary layer growth near trailing edge (1.15C ax streamwise location) for (a) Case II: constant fillet, (b) Case III: Variable fillet.

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Fig. 18. Velocity vector plot imposed with relative total pressure loss coefficient (a) base case, (b) 1-20-LTE, (c) 2-20-LTE, (d) 3-20-LTE, (e) 4-20-LTE, (f) 5-20-LTE.

blade. Fig. 18 shows the horse shoe vortex formation superimposed with rotor total-pressure loss coefficient (RLC) in a stagnation-line plane. Confinement of higher loss coefficient is seen within the vortex core region. As the leading edge radius increases (Fig. 18), reduction in the intensity as well as the size of the horse shoe vortex is seen when compared to the base case. Horse shoe vortex almost disappears for 5 mm radius fillet. This reduction comes from the concave nature of the fillet surface as it provides the local acceleration to the flow. This reduces the strength of adverse pressure gradient, the main driving force for horse-shoe vortex formation, imposed by the blade leading edge. It results in the movement of the stagnation point downstream close to leading

edge i.e. taking lesser portion of flow in the vortex system. Further, it leads to a reduction in the shape of the passage vortex, which will be discussed in the next section. For a 1 mm fillet case, a sharp increase in the concentration of loss coefficient at the vortex core is seen. This may be due to the fact that a fillet shape is felt only by the separated fluid, i.e. saddle point position at which the boundary layer fluid gets separated from the endwall is not affected by it. Overall, a view of the fillet effectiveness over the total pressure loss reduction can be interpreted from the plot of mass averaged relative total pressure loss coefficient at the rotor exit (Fig. 19). At the top, a comparison between constant and variable fillet cases for

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ation in included angle was insignificant compared to the leading edge radii variation. The effectiveness of variable fillet case 5-20-LE in reducing the strength and transformation of horseshoe vortex into passage vortex near hub is shown in Fig. 20. In order to make evident this compared to baseline case, relative total pressure loss coefficient (RLC) is plotted at different streamwise planes (0.15C ax , 0.35C ax , 0.65C ax , 0.85C ax , 0.95C ax ) in rotor blade passage. Improvement in both aspects like reduction in intensity as well as the size are observed in each streamwise plane for Case III: variable fillet case. Further corner vortices associated with the base case got diminished due to filleting. 5. Conclusions Fillet designs near vane/blade endwall juncture are proposed, which effectively modify the secondary flow field. Constant & variable fillet designs were investigated by varying the radius and the included angle at leading edge and trailing edge regions. The conclusions derived from the present numerical studies were: Fig. 19. Comparison of relative total pressure loss coefficient (RLC) between constant and variable radii fillet with base case having no fillet application.

different included angles is shown, compared to baseline. It was clearly observed that the losses associated with the constant fillet cases were significantly high (above 3.5%) irrespective of the included angle variation. Thick boundary layer growth after the throat is the reason for this, as mentioned in the previous section. Reducing the included angle (θ ) reduces the TKE associated with the vortex structures. Nevertheless this effect is not reflected in the total pressure loss coefficient at the exit. An included angle of 20 deg shows the variation at the most of 0.5%. In the second plot, it is shown for the various leading edge radius cases. Above 4 mm fillet radii, change in trend of total pressure variation occurred. This is due to the fact that the variation of fillet shape on suction and pressure side along the stream-wise direction played a role in reducing the passage vortex formation. Maximum amount of reduction was obtained for the 5 mm radii case and the vari-

• For NGV, an increase in the leading edge fillet radius exhibits an increment in three dimensional region along the span. However, there is no significant effect on included angle variation. • Variable fillet having stream-wise radial variation performs better, as it reduces the height and strength of secondary flow regime along the span, when compared to constant fillet. Further, it helps in overcoming the separation bubble near the trailing edge of NGV. • For rotor case also variable fillet works better as it is reduces the losses by minimizing the boundary layer growth near the trailing edge when compared with constant fillet. Further, as the leading edge radius of variable fillet increases, there is a reduction in the intensity and size of horse shoe vortex, and thus secondary flow losses. • Fillet cases 1-20-LE for NGV and 5-20-LE case for rotor shows the best improvement in total pressure loss coefficient along with the span-wise exit angle distribution.

Fig. 20. Pressure side leg of Horseshoe vortex mapped with turbulent kinetic energy values along stream-wise locations (from left) 0.15C ax , 0.35C ax , 0.65C ax , 0.85C ax , 0.95C ax for (a) base case, (b) 5-20-LE.

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