TURBINE BUCKET FOR USE IN GAS TURBINE ENGINES AND METHODS FOR FABRICATING THE SAME

Abstract

A turbine bucket for use with a turbine engine. The turbine bucket includes a dovetail that is coupled to a rotor assembly that is positioned within a turbine casing. A platform extends from the dovetail. An airfoil extends from the platform. The airfoil includes a root end and a tip end. The tip end extends outwardly from the root end towards the turbine casing. A tip shroud extends from the tip end. The tip shroud includes a shroud plate. A first shroud rail extending a first radial distance from the shroud plate towards the turbine casing. A second shroud rail extends a second radial distance from the shroud plate towards the turbine casing that is different than the first radial distance.

Description

BACKGROUND OF THE INVENTION

The field of the disclosure relate generally to gas turbine engines, and more particularly, to a turbine bucket used with gas turbine engines.

At least some known gas turbine engines include a combustor, a compressor coupled downstream from the combustor, a turbine, and a rotor assembly rotatably coupled between the compressor and the turbine. Some known rotor assemblies include a rotor shaft, at least one rotor disk coupled to the rotor shaft, and a plurality of circumferentially-spaced turbine buckets that extend outward from each rotor disk. Each turbine bucket includes an airfoil that extends radially outward from a platform towards a turbine casing.

During operation of at least some known turbines, the compressor compresses air that is subsequently mixed with fuel prior to being channeled to the combustor. The mixture is then ignited generating hot combustion gases that are then channeled to the turbine. The rotating turbine blades or buckets channel high-temperature fluids, such as combustion gases, through the turbine. The turbine extracts energy from the combustion gases for powering the compressor, as well as producing useful work to power a load, such as an electrical generator, or to propel an aircraft in flight.

At least some known turbine buckets include a shroud that extends from an outer tip end of the airfoil to reduce air flow between the airfoil and the turbine casing. At least a portion of combustion gases channeled through the turbine are undesirably channeled between the tip shroud and the airfoil as tip clearance losses. Such tip clearance loses may constitute between approximately 20-25% of the total losses in a turbine stage. Known tip shrouds include a cavity that is defined between the leading edge of the turbine bucket and the turbine casing. The cavity traps combustion gases and causes a vortex to form within the cavity. The vortex redirects combustion gases towards the main flow path and disrupts combustion gases within the main flow path. However, this disruption generates secondary flow losses within the main gas path that reduce an operating efficiency of the turbine.

BRIEF DESCRIPTION OF THE INVENTION

In one aspect, a turbine bucket for use with a turbine engine is provided. The turbine bucket includes a dovetail that is coupled to a rotor assembly that is positioned within a turbine casing. A platform extends from the dovetail. An airfoil extends from the platform. The airfoil includes a root end and a tip end. The tip end extends outwardly from the root end towards the turbine casing. A tip shroud extends from the tip end. The tip shroud includes a shroud plate. A first shroud rail extending a first radial distance from the shroud plate towards the turbine casing. A second shroud rail extends a second radial distance from the shroud plate towards the turbine casing that is different than the first radial distance.

In another aspect, a turbine engine system is provide. The turbine engine system includes a casing, a compressor, and a turbine that is coupled in flow communication with the compressor to receive at least some of the air discharged by the compressor. The turbine is positioned within the casing. A rotor shaft is rotatably coupled to the turbine. The rotor shaft defines a centerline axis. A plurality of circumferentially-spaced turbine buckets are coupled to the rotor shaft. Each of the plurality of turbine buckets includes a platform. An airfoil extends from the platform. The airfoil includes a root end and a tip end. The tip end extends outwardly from the root end towards the casing. A tip shroud extends from the tip end. The tip shroud includes a shroud plate. A first shroud rail extends a first radial distance from the shroud plate towards the casing. A second shroud rail extends a second radial distance from the shroud plate towards the casing that is different than the first radial distance.

In a further aspect, a method for fabricating a turbine bucket for use with a turbine engine is provided. The method includes forming a substantially solid ceramic turbine bucket core. The core is inserted into a die. The turbine bucket is cast with an airfoil extending from a platform. Wherein the airfoil includes a tip shroud that extends from a tip end of the airfoil. The tip shroud includes a shroud plate, a first shroud rail that extends a first radial distance from the shroud plate, and a second shroud rail that extends a second radial distance from the shroud plate that is different than the first radial distance.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic view of an exemplary turbine engine.

FIG. 2 is a partial sectional view of a portion of an exemplary rotor assembly that may be used with the gas turbine engine shown in FIG. 1.

FIG. 3 is an enlarged partial sectional view of a portion of the rotor assembly shown in FIG. 2 and taken along area 3.

FIG. 4 is a partial perspective view of the rotor assembly shown in FIG. 3.

FIG. 5 is a partial sectional view of an alternative rotor assembly that may be used with the gas turbine engine shown in FIG. 1.

DETAILED DESCRIPTION OF THE INVENTION

The exemplary methods and systems described herein overcome at least some disadvantages of known turbine buckets by providing a tip shroud that facilitates reducing the formation of vortices near the leading edge of the turbine bucket. More specifically, the embodiments described herein provide a tip shroud that includes a plurality of shroud rails that have radial heights that reduce the size of a cavity defined between the tip shroud and a turbine casing such that the formation of vortices within the cavity are reduced.

As used herein, the term “upstream” refers to a forward or inlet end of a gas turbine engine, and the term “downstream” refers to an aft or nozzle end of the gas turbine engine.

FIG. 1 is a schematic view of an exemplary turbine engine system 10. In the exemplary embodiment, turbine engine system 10 includes an intake section 12, a compressor section 14 that is coupled downstream from intake section 12, a combustor section 16 that is coupled downstream from compressor section 14, a turbine section 18 that is coupled downstream from combustor section 16, and an exhaust section 20 that is coupled to turbine section 18. Turbine section 18 is coupled to compressor section 14 via a rotor shaft 22. In the exemplary embodiment, combustor section 16 includes a plurality of combustors 24. Combustor section 16 is coupled to compressor section 14 such that each combustor 24 is in flow communication with the compressor section 14. Turbine section 18 is coupled to compressor section 14 and to a load 26 such as, but not limited to, an electrical generator and/or a mechanical drive application. In the exemplary embodiment, each compressor section 14 and turbine section 18 includes at least one rotor assembly 28 that is coupled to rotor shaft 22.

During operation, intake section 12 channels air towards compressor section 14 wherein the air is compressed to a higher pressure and temperature prior to being discharged towards combustor section 16. Combustor section 16 mixes the compressed air with fuel, ignites the fuel-air mixture to generate combustion gases, and channels the combustion gases towards turbine section 18. More specifically, in combustors 24, fuel, for example, natural gas and/or fuel oil, is injected into the air flow, and the fuel-air mixture is ignited to generate high temperature combustion gases that are channeled towards turbine section 18. Turbine section 18 converts thermal energy from the gas stream to mechanical rotational energy as the combustion gases impart rotational energy to turbine section 18 and to rotor assembly 28.

FIG. 2 is a partial sectional view of a portion of rotor assembly 28 that may be used with turbine engine system 10. FIG. 3 is an enlarged partial sectional view of rotor assembly 28 taken along area 3. FIG. 4 is a partial perspective view of rotor assembly 28. In the exemplary embodiment, turbine section 18 includes a plurality of stages 30 that each include a stationary row of stator vanes 32 and a row of rotating turbine buckets 34. Turbine buckets 34 each extend radially outward from a rotor disk 36. Each rotor disk 36 is coupled to rotor shaft 22 and rotates about a centerline axis 38 that is defined by rotor shaft 22. A turbine casing 40 extends circumferentially about rotor assembly 28 and stator vanes 32. Stator vanes 32 are each coupled to casing 40 and extend radially inward from casing 40 towards rotor shaft 22.

In the exemplary embodiment, each rotor disk 36 is annular and includes a central bore 42 that extends substantially axially therethrough. More specifically, a disk body 44 extends radially outwardly from central bore 42 and is oriented substantially perpendicularly to centerline axis 38. Central bore 42 is sized to receive rotor shaft 22 therethrough. Disk body 44 extends radially between a radially inner edge 46 and a radially outer edge 48, and axially from an upstream surface 50 to an opposite downstream surface 52. Upstream surface 50 and downstream surface 52 each extend between inner edge 46 and outer edge 48. A support arm 54 is coupled between adjacent rotor disks 36 to form rotor assembly 28.

Each turbine bucket 34 is coupled to disk outer edge 48 and is spaced circumferentially about rotor disk 36. Adjacent rotor disks 36 are oriented such that a gap 56 is defined between each row 58 of circumferentially-spaced turbine buckets 34. Gap 56 is sized to receive a row 60 of stator vanes 32 that are spaced circumferentially about rotor shaft 22. Stator vanes 32 are oriented to channel combustion gases downstream towards turbine buckets 34. A combustion gas path 62 is defined between turbine casing 40 and each rotor disk 36. Each row 58 and 60 of turbine buckets 34 and stator vanes 32 extends at least partially through a portion of combustion gas path 62.

In the exemplary embodiment, each turbine bucket 34 extends radially outwardly from disk body 44 and each includes an airfoil 64, a tip shroud 66, a platform 68, a shank 70, and a dovetail 72. Airfoil 64 extends generally radially between platform 68 and tip shroud 66. Platform 68 extends between airfoil 64 and shank 70 such that each airfoil 64 extends radially outwardly from platform 68 towards turbine casing 40. Shank 70 extends radially inwardly from platform 68 to dovetail 72. Dovetail 72 extends radially inwardly from shank 70 and enables turbine buckets 34 to securely couple to rotor disk 36. Shank 70 includes a forward cover plate 74 and an opposite aft cover plate 76.

In the exemplary embodiment, a forward angel wing 78 extends outwardly from forward cover plate 74 to facilitate sealing a forward buffer cavity 80 defined between rotor disk upstream surface 50 and stator vane 32. An aft angel wing 82 extends outwardly from aft cover plate 76 to facilitate sealing an aft buffer cavity 84 defined between rotor disk downstream surface 52 and stator vane 32. In the exemplary embodiment, a forward lower angel wing 86 extends outwardly from forward cover plate 74 to facilitate sealing between turbine bucket 34 and rotor disk 36. More specifically, forward lower angel wing 86 is positioned between dovetail 72 and forward angel wing 78.

In the exemplary embodiment, airfoil 64 extends radially between a root end 88 and a tip end 90 and includes a radial length 92 that is defined between root end 88 and tip end 90. Root end 88 is adjacent to platform 68. Airfoil 64 extends radially outwardly from platform 68 towards turbine casing 40 such that tip end 90 is positioned adjacent turbine casing 40. In the exemplary embodiment, airfoil 64 has a pressure side 94 and a suction side 96. Each side 94 and 96 extends generally axially between a leading edge 98 and a trailing edge 100. Pressure side 94 is generally concave and suction side 96 is generally convex. In the exemplary embodiment, airfoil 64 has an axial width 102 defined between leading edge 98 and trailing edge 100. In one embodiment, tip end 90 has a first axial width 104, and root end 88 has a second axial width 106 that is longer than first axial width 104.

In the exemplary embodiment, tip shroud 66 extends from tip end 90 of airfoil 64 and between tip end 90 and turbine casing 40. Tip shroud 66 includes a plurality of shroud rails 108 that extend from a shroud plate 110. In one embodiment, shroud rails 108 are coupled to shroud plate 110. In an alternative embodiment, shroud rails 108 are formed integrally with shroud plate 110. In the exemplary embodiment, shroud plate 110 is substantially rectangular and extends between a leading surface 112 and an opposite trailing surface 114 and between a first outer edge 116 and an opposite circumferentially-spaced second outer edge 118. A Z-notch groove 120 is defined in each first outer edge 116 and second outer edge 118 to facilitate coupling shroud plate 110 to an adjacent shroud plate 110. In the exemplary embodiment, shroud plate 110 has a circumferential width 122 defined between edges 116 and 118. Shroud plate 110 also has an axial length 124 defined between surfaces 112 and 114. In the exemplary embodiment, axial length 124 is approximately equal to axial width 104 of tip end 90. Alternatively, shroud plate 110 may have an axial length 124 that is longer than, or shorter than, axial width 104.

In the exemplary embodiment, each shroud rail 108 includes a sidewall 126 that includes an upstream surface 128 and a downstream surface 130. Shroud rail 108 has a circumferential width 132 defined between plate edges 116 and 118. Each sidewall 126 extends generally radially between a radially outer surface 134 and a radially inner surface 136, and has a radial height 138 defined between surfaces 136 and 134. Each surface 128 and 130 extends between surfaces 136 and 134. Radially inner surface 136 extends from an outer surface 140 of shroud plate 110. Sidewall 126 extends a radial distance 142 towards turbine casing 40 from outer surface 140 to radially outer surface 134.

In the exemplary embodiment, turbine casing 40 includes an inner surface 144 that circumscribes rotor assembly 28. Inner surface 144 includes a layer 146 of abradable material. Shroud rail 108 is positioned adjacent to inner surface 144 such that radially outer surface 134 contacts at least a portion of abradable layer 146 such that a portion of abradable layer 146 is removed during rotation of rotor assembly 28 as turbine bucket 34 thermally expands. In one embodiment, shroud rail 108 includes at least one cutting tooth 148 that extends outwardly from surfaces 128 and 130. Each cutting tooth 148 facilitates removing abradable layer 146 during rotation of rotor assembly 28.

In the exemplary embodiment, tip shroud 66 includes a first shroud rail 150 and a second shroud rail 152. First shroud rail 150 is positioned closer to leading surface 112 than second shroud rail 152. First shroud rail 150 is positioned adjacent to leading surface 112 such that a first cavity 154, i.e. a forward cavity, is defined between casing inner surface 144 and each upstream surface 128. Second shroud rail 152 is spaced axially from first shroud rail 150 along centerline axis 38 such that a second cavity 156, i.e. an interior cavity, is defined between inner surface 144, plate outer surface 140, rail downstream surface 130 and rail upstream surface 128. Second shroud rail 152 is positioned with respect trailing surface 114 such that a third cavity 158, i.e. an aft cavity, is defined between downstream surface 130 of second shroud rail 152, plate outer surface 140, and inner surface 144. In the exemplary embodiment, rails 150 and 152 are each adjacent to casing inner surface 144 such that a tip fluid flow path 160 is defined between tip shroud 66 and inner surface 144. Tip fluid flow path 160 channels at least a portion of the combustion gases channeled through rotor assembly 28 between tip shroud 66 and turbine casing 40.

In the exemplary embodiment, first shroud rail 150 extends a first radial distance 162 between plate outer surface 140 and rail outer surface 134. Second shroud rail 152 extends a second radial distance 164 between outer surface 140 and rail outer surface 134. In the exemplary embodiment, first radial distance 162 is different than second radial distance 164. For example, in one embodiment, first radial distance 162 is shorter than second radial distance 164 of second shroud rail 152. In another alternative embodiment, first radial distance 162 is between about 40% and about 60% of second radial distance 164.

In the exemplary embodiment, turbine casing 40 includes a first abradable surface 166 adjacent to first shroud rail 150, and a second abradable surface 168 adjacent to second shroud rail 152. First abradable surface 166 and second abradable surface 168 are each substantially parallel to centerline axis 38. In the exemplary embodiment, radially outer surfaces 134 of rails 150 and 152 are each substantially parallel to centerline axis 38. Alternatively, outer surfaces 134 may be oriented obliquely with respect to centerline axis 38.

In the exemplary embodiment, a tangential rail surface 170 is defined between radially outer surfaces 134 of rails 150 and 152. A shroud surface plane 172 is defined along plate outer surface 140 from leading surface 112 to trailing surface 114. A centerline plane 174 is defined parallel to shaft centerline axis 38. In the exemplary embodiment, first shroud rail 150 is sized and oriented with respect to second shroud plate 110 such that tangential rail surface 170 extends obliquely with respect to shroud surface 172 at a first angle α₁ defined between tangential rail surface 170 and shroud surface 172. Shroud plate 110 is oriented obliquely with respect to centerline plane 174 such that a second oblique angle α₂ is defined between shroud surface plane 172 and centerline plane 174. In one embodiment, first oblique angle α₁ is greater than second oblique angle α₂. In another alternative embodiment, first oblique angle α₁ is less than second oblique angle α₂.

During operation, compressor section 14 (shown in FIG. 1) compresses air and discharges compressed air into combustor section 16 (shown in FIG. 1) and towards turbine section 18. The majority of air discharged from compressor section 14 is channeled towards combustor section 16. More specifically, pressurized compressed air is channeled to combustors 24 (shown in FIG. 1) wherein the air is mixed with fuel and ignited to generate high temperature combustion gases 176. The combustion gases 176 are channeled towards combustion gas path 62, wherein the gases 176 impinge upon turbine buckets 34 and stator vanes 32 to facilitate imparting a rotational force on rotor assembly 28. At least a portion of combustion gases 176 impinge turbine buckets 34, is directed into tip fluid flow path 160, and is channeled between tip shroud 66 and turbine casing 40. As combustion gases 176 flow through tip fluid flow path 160, first shroud rail 150 is sized and oriented to define a size of forward cavity 154 to facilitate reducing the formation of vortices along tip fluid flow path 160, and to reduce fluid interference between combustion gas path 62 and tip fluid flow path 160.

FIG. 5 is a sectional view of an alternative embodiment of rotor assembly 28 shown in FIG. 3. Identical components shown in FIG. 5 are labeled with the same reference numbers used in FIG. 3. In an alternative embodiment, first shroud rail 150 includes a first rail distance 162 that is longer than a second rail distance 164 of second shroud rail 152. In one embodiment, first shroud rail 150 is sized with respect to second shroud rail 152 such that tangential rail surface 170 is substantially parallel to centerline axis 38. In such an embodiment, first abradable surface 166 and second abradable surface 168 are each substantially parallel to centerline axis and are each positioned an approximately equal radial distance 178 from centerline plane 174. In this embodiment, second shroud rail 152 defines an aft cavity 158 that is sized, shaped, and oriented to facilitate reducing a formation of vortices along tip fluid flow path 160, and to reduce fluid interference between combustion gas path 62 and tip fluid flow path 160.

In one embodiment, turbine buckets 34 are fabricated by casting a core (not shown). The core is fabricated by injecting a liquid ceramic and graphite slurry into a core die (not shown), and the slurry is heated to form a solid ceramic blade core. The blade core is suspended in an blade die (not shown) and hot wax is injected into the blade die to surround the ceramic blade core. The hot wax solidifies and forms a wax blade with the ceramic core suspended in the blade.

The wax blade with the ceramic core is repeatedly dipped into ceramic slurry to form a ceramic shell outside the wax blade. The core, wax, and shell cluster is then heated to an elevated temperature to remove the wax and form a casting mold with ceramic core in the middle. The molten metal is then poured into the hollow casting mold. The molten metal takes the place of the wax blade, and forms a metal turbine bucket with the ceramic core remaining in place. The turbine bucket is then cooled, and the ceramic core removed.

The above-described turbine bucket overcome at least some disadvantages of known turbine buckets by reducing the formation of vortices within a combustion gas path between the turbine bucket and a turbine casing. More specifically, by providing a tip shroud that includes a plurality of shroud rails having different radial heights, the size of a cavity defined between the turbine bucket and the turbine casing is reduced. By reducing the size of the cavity, the formation of vortices that redirect combustion gases towards a main flow path defined between airfoils is reduced. Moreover, the secondary flow losses that are generated within the main gas path are reduced, thus reducing the losses in gas energy and increasing the useful life of the turbine engine.

Exemplary embodiments of a turbine bucket for use in a gas turbine engine and method for assembling the same are described above in detail. The methods and apparatus are not limited to the specific embodiments described herein, but rather, components of systems and/or steps of the method may be utilized independently and separately from other components and/or steps described herein. For example, the methods and apparatus may also be used in combination with other combustion systems and methods, and are not limited to practice with only the gas turbine engine assembly as described herein. Rather, the exemplary embodiment can be implemented and utilized in connection with many other combustion system applications.

Although specific features of various embodiments of the invention may be shown in some drawings and not in others, this is for convenience only. Moreover, references to “one embodiment” in the above description are not intended to be interpreted as excluding the existence of additional embodiments that also incorporate the recited features. In accordance with the principles of the invention, any feature of a drawing may be referenced and/or claimed in combination with any feature of any other drawing.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.

Claims

1. A turbine bucket for use with a turbine engine, said turbine bucket comprising:

a dovetail coupled to a rotor assembly positioned within a turbine casing;

a platform extending from said dovetail;

an airfoil extending from said platform, said airfoil comprising a root end and a tip end, said tip end extending outwardly from said root end towards the turbine casing; and

a tip shroud extending from said tip end, said tip shroud comprising: a shroud plate; a first shroud rail extending a first radial distance from said shroud plate towards the turbine casing; and a second shroud rail extending a second radial distance from said shroud plate towards the turbine casing that is different than said first radial distance.

2. A turbine bucket in accordance with claim 1, wherein said shroud plate extends generally between a leading surface and a trailing surface, said first shroud rail positioned closer to said leading surface than said second shroud rail such that an axial flow path is defined between the turbine casing and said tip shroud.

3. A turbine bucket in accordance with claim 2, wherein said first radial distance is longer than said second radial distance.

4. A turbine bucket in accordance with claim 2, wherein said first radial distance is shorter than said second radial distance.

5. A turbine bucket in accordance with claim 1, wherein said first radial distance is between about 40% to 60% of said second radial distance.

6. A turbine bucket in accordance with claim 3, wherein said first shroud rail is positioned a distance from said leading surface such that a cavity is defined between said leading surface and the turbine casing, said cavity facilitates reducing a formation of vortices within said cavity.

7. A turbine bucket in accordance with claim 4, wherein said second shroud rail is positioned a distance from said trailing surface such that a cavity is defined between said trailing surface and the turbine casing, said cavity facilitates reducing a formation of vortices within said cavity.

8. A turbine bucket in accordance with claim 1, wherein said first and second shroud rails define an tangential rail surface that is oriented at an oblique angle with respect to an outer surface of said shroud plate.

9. A turbine engine system comprising:

a casing;

a compressor;

a turbine coupled in flow communication with said compressor to receive at least some of the air discharged by said compressor, said turbine positioned within said casing;

a rotor shaft rotatably coupled to said turbine, said rotor shaft defining a centerline axis; and

a plurality of circumferentially-spaced turbine buckets coupled to said rotor shaft, each of said plurality of turbine buckets comprising: a platform; an airfoil extending from said platform, said airfoil comprising a root end and a tip end, said tip end extending outwardly from said root end towards said casing; and a tip shroud extending from said tip end, said tip shroud comprising: a shroud plate; a first shroud rail extending a first radial distance from said shroud plate towards said casing; and a second shroud rail extending a second radial distance from said shroud plate towards said casing that is different than said first radial distance.

10. A turbine engine system in accordance with claim 9, wherein said shroud plate extends generally between a leading surface and a trailing surface, said first shroud rail positioned closer to said leading surface than said second shroud rail such that an axial flow path is defined between said casing and said tip shroud.

11. A turbine engine system in accordance with claim 10, wherein said first radial distance is longer than said second radial distance.

12. A turbine engine system in accordance with claim 10, wherein said first radial distance is shorter than said second radial distance.

13. A turbine engine system in accordance with claim 9, wherein said first radial distance is between about 40% to 60% of said second radial distance.

14. A turbine engine system in accordance with claim 11, wherein said first shroud rail is positioned a distance from said leading surface such that a cavity is defined between said leading surface and said casing, said cavity facilitates reducing a formation of vortices within said cavity.

15. A turbine engine system in accordance with claim 12, wherein said second shroud rail is positioned a distance from said trailing surface such that a cavity is defined between said trailing surface and said casing, said cavity facilitates reducing a formation of vortices within said cavity.

16. A method for fabricating a turbine bucket for use with a turbine engine, said method comprising:

forming a substantially solid ceramic turbine bucket core;

inserting the core into a die; and

casting the turbine bucket with an airfoil extend from a platform, wherein the airfoil includes a tip shroud extending from a tip end of the airfoil, the tip shroud including a shroud plate, a first shroud rail extending a first radial distance from the shroud plate, and a second shroud rail extending a second radial distance from the shroud plate that is different than the first radial distance.

17. A method in accordance with claim 16, wherein casting the turbine bucket further comprises casting the turbine bucket such that the first shroud rail is positioned closer to a leading surface of the shroud plate than the second shroud rail such that an axial flow path is defined from the first shroud rail to the second shroud rail.

18. A method in accordance with claim 17, wherein casting the turbine bucket further comprises casting the turbine bucket such that the first radial distance is longer than the second radial distance.

19. A method in accordance with claim 17, wherein casting the turbine bucket further comprises casting the turbine bucket such that the first radial distance is shorter than said second radial distance.

20. A method in accordance with claim 16, wherein casting the turbine bucket further comprises casting the turbine bucket such that the first radial distance is between about 40% to 60% of the second radial distance.