Parametric 3D Blade Geometry Modeling Tool for. Turbomachinery Systems

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2 Parametric 3D Blade Geometry Modeling Tool for Turbomachinery Systems A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in the School of Aerospace Systems of the College of Engineering and Applied Science by Kiran Siddappaji B.Tech, Maulana Azad National Institute of Technology, India, 2008 Committee Chair: Dr. Mark G. Turner

3 Abstract Turbomachinery blades are an integral part of air breathing propulsion systems, gas and steam turbines and other energy conversion devices. The blade design is a very important process since it defines component performance. A parametric approach for the blade geometry design has been implemented. A variety of three dimensional blade shapes can be created using only a few basic parameters and limited interaction with a CAD system. Using a general approach for creating the blade geometries makes the process easy and robust for creating 3D blade shapes for various turbomachinery components. The geometry of the blade is defined by a very basic set of geometric and aerodynamic parameters. Parameters such as flow angles, axial chord, thickness to chord ratio and streamline meridional coordinates are defined. The leading edge and trailing edge are defined by curves as part of the input. Using these parameters, a specified number of 2D airfoils are created and are radially stacked on the desired stacking axis. The sweep and lean perturbations of the blade are defined by splines as a function of a few control points. The design tool generates a specified number of 3D blade sections and each section consists of a defined number of coordinates in the cartesian coordinate system. These sections can be lofted in a CAD package to obtain a solid 3D blade model, which has been demonstrated using Unigraphics-NX. Parametric update of the spline points defining the 3D blade sections creates new blade shapes without going back into the CAD interface. This approach for the design is very beneficial as the geometry can be modified quickly and easily as per the needs of the designer at any point of time. Using this tool, blade shapes of a 10 stage compressor similar to the GE/NASA EEE HPC, a 3 stage booster, a reverse engineered GE 1.5 MW wind turbine and a centrifugal compressor based on a NASA design are constructed as examples. The general capabilty of the design tool is demonstrated through these examples. i

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5 Acknowledgments The author is grateful to NASA for funding and support through NRA project- "Advanced Design Techniques for MDAO of Turbomachinery with Emphasis for the Engine System". The MDAO project meetings with the rest of the members proved to be very helpful in terms of their feedback. Rob Ogden deserves a special thanks for his system related support and the members of the UC Gas Turbine Simulation Laboratory for their assistance and clearing out certain misconcepts. Thanks to David Gutzwiller and Kevin Park for taking time out of their schedule to provide necessary input and helping in debugging the code several times. Sincere thanks to Dr. Ali Merchant for providing assistance and guidance at odd hours inspite of his hectic schedule. The preliminary version of the blade geometry generator was part of T-Axi that Dr. Merchant wrote. For that also, the author is grateful. Author is grateful to Dr. Dario Bruna for providing clear suggestions in the initial phase. Thanks are due to Istvan Iszabo and Soumitr Dey for giving very valuable lessons and tips in UG/NX which saved a lot of modeling time. The author would like to thank his thesis advisor, Dr. Mark G. Turner, for his insight, guidance and feedback. His methods of tackling an issue and solving it in a simple way is worth the mention. Lastly, a special thanks to my parents and friends for providing me the constant support and encouragement. iii

6 Contents Abstract i Acknowledgments iii Contents vi List of Figures ix Nomenclature x 1 Introduction Motivation Literature Review and Previous Work Goals and Thesis Layout D Blade Generator Input File Governing Equations Blade section construction Variety of Airfoil shapes Leading Edge and Trailing Edge Curves Streamwise Mapping of the Blade Sections Sweep and Lean to the Blade Construction of Uniform B-Splines Radial Stacking of 3D Blade Sections iv

7 2.8 Coordinate Transformation Output files D Blade CAD Model Connecting with CAPRI Extruded Blade CFD Analysis of the 3D blade Gridding CFD solver: Euranus Structural Analysis of the 3D Blade 36 5 Example Design Problems Stage EEE High Pressure Compressor Design Free vortex Design Full Definition 10 stage Design Stage Booster Design for Turbofan Engine Reverse-Engineered Wind Turbine Design Low-Speed Centrifugal Compressor design Conclusion and Future Work Conclusion Future Work References 60 Appendix 62 A Input Files for Example Cases 63 A.1 Rotor 3 of 10 Stage EEE HPC Free Vortex Design A.2 Rotor 3 of 10 Stage EEE HPC Full Definition Design A.2.1 Blade Metal Angles from the NASA Report [1] A.3 Rotor 3 of 3 Stage Booster Design v

8 A.4 Reverse Engineered Wind Turbine Design A.5 Low Speed Centrifugal Compressor Design vi

9 List of Figures 2.1 3D Blade Design Process Flowchart r x θ space for the 3D blade Meridional view of the blade showing axisymmetric streamlines as construction curves A blade showing the blade sections Blade section on the mean camber curve Relationship between (m b, θ b) system and (u, v) system Plot of the camber line of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design Camber angle plot of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design Plot showing the curvature of the camber line of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design Thickness distribution over a non-dimensional unit camber length [2] Blade section on the normalized unit chord Variety of airfoils generated by 3DBGB in addition to circular and 4 digit NACA airfoils Stream surface for an axial machine and a radial machine respectively m sle obtained using inverse spline on m s x 3D obtained by spline evaluation at each m 3D of each blade section r 3D obtained by spline evaluation at each m 3D of each blade section Mapping of the blade section to the corresponding stream surface vii

10 2.18 δm evaluated at each span Forward and backward swept blades respectively Spanwise lean modification on rotor 3 of a booster optimum resulting in lean near the hub Construction of a uniform non-rational B-spline curve. [3] θ offset defined for various stacking option D Blade lofted in UG Offset at the hub streamline Extruded blade at the tip with hub and casing Creating the meridional and blade to blade mesh Characteristic curve showing Total PR vs mass flow rate Characterisitic curve showing Isentropic efficiency vs mass flow rate Relative Mach contours of rotor 3 of the optimized booster with stream ribbons at over the tip clearance Spanwise lean modification on rotor Iso surface comparison of axial velocity for original and modified rotor 3 of the booster optimum point 2 [4] Blade meshed in ANSYS V Contour plots of Displacement vector sum for first 7 modes of a blade under Nodal solution Contour plots of Displacement vector sum for last 7 modes of a blade under Nodal solution Displaced blade E 3 10 Stage Compressor Cross Section. [1] Axisymmetric grid showing the streamlines of 10 Stage HPC free vortex design and the straight leading and trailing edge locations of the blade rows in green Rotor 1 and rotor 3 of EEE HPC free vortex design created using 3DBGB Rotor 1 and Rotor 3 blisk assembly for EEE HPC free vortex design viii

11 5.5 Axisymmetric grid showing the streamlines of E 3 10 Stage HPC full definition design and the leading and trailing edge locations of the blade rows in green Comparison of blade metal angles of EEE-HPC rotor 3 given in the report [1] and generated by 3DBGB Rotor 3 of GE EEE HPC full definition design with non-straight leading and trailing edges Rotor and stator assembly for 10 stage HPC full definition design from the GE/E 3 parameters stage booster of a GE90 turbofan engine.[5] Axisymmetric grid of the 3-stage optimized booster showing the streamlines and the leading and trailing edge locations of the blade rows in green stage booster rotor assembly with the fan blades stage booster assembly with split casing Tetrahedral mesh of rotor 3 of a 3 stage booster in ANSYS Contour plots of displacement vector sum of all 14 modes of rotor 3 under Modal solution GE 1.5 MW wind turbine blade. [6] Axisymmetric grid of the reverse engineered GE 1.5 MW wind turbine blade Reverse engineered GE 1.5 MW wind turbine blade geometry created using 3DBGB Reverse Engineered GE 1.5 MW wind turbine assembly Meshed wind-turbine blade with 8 node Hexahedron Brick 185 element Von-Mises Stress plot on GE reverse engineered blade First 5 Modal solution of GE reverse engineered blade LSCC flowpath with actual measurements. [7] A radial compressor flowpath showing the hub and casing with streamlines and the leading and trailing edge locations Velocity triangle at the leading edge for the radial compressor Radial compressor model compared with NASA LSCC [7] ix

12 Nomenclature a b i L m mm m np nspn r u v N U V x Offset value Intermediate extruded blade scale factor Incidence angle Reference length Meridional coordinate Millimeters Normalized Meridional coordinate Number of points Number of blade sections Radial coordinate Non-dimensional chordwise coordinate Non-dimensional coordinate perpendicular to u Rotational Speed Wheel Speed Absolute flow velocity Axial coordinate Greek α β β δ Absolute flow angle Relative flow angle Blade metal angle Deviation angle x

13 ζ θ Stagger angle Tangential direction π φ Meridional plane flow angle = tan 1 (V r /V x ) ω Loss coefficient Subscripts b bbot bstgr btop r ref x s t z LE ND Norm T E 3D Blade Blade bottom Blade stagger Blade top Radial Reference Axial Streamline Tip Axial Leading Edge Non Dimensional Normal Trailing Edge Three dimensional Units Hz m/s MPa MW Hertz meters per second Mega Pascals Mega Watts xi

14 Abbreviations 2D 3D 3DBGB CAD CAPRI CFD EEE FEA GUI IGES IGV NACA NURBS RE RPM UG Two Dimensional Three Dimensional Three Dimensional Blade Geometry Builder Computer Aided Design Computational Analysis PRogramming Interface Computational Fluid Dynamics Energy Efficient Engine Finite Element Analysis Graphic User Interface Initial Graphics Exchange Specification Inlet Guide Vane National Advisory Committee for Aeronautics Non Uniform Rational B-Spline Reverse Engineered Rotations Per Minute UniGraphics-NX xii

15 Chapter 1 Introduction 1.1 Motivation The gas turbine blade design process has become more challenging recently as the approach has become more parametric and innovative. However, the basic philosophy of any blade design still remains the same in terms of building the 2D sections utilizing all the necessary geometric parameters and then constructing the 3D model. There are many software tools available which are built based on the user needs. The attempt here is to build a blade design tool based on very few but essential parameters both geometric and aerodynamic, which can be automated and be a part of any optimization chain and above all be flexible to create a wide range of geometries with minimal parametric modifications. The 3D model is created using the 3D blade sections output from the design tool and the final blade model is obtained by the limited interaction with a CAD package. Subsequent blade models can be generated by parametric spline updates on the base model created using the CAD package. The 3D blade shape is ready for validation through 3D CFD analysis and FEA analysis. The geometry can be quickly modified by just changing the parameters and the process is continued without losing time in remodeling. There are many 3D blade design packages which are either completely proprietary or commercial implementations of CAD based models. The tool presented here has general capability of designing various turbomachinery blade shapes and the parametric approach is beneficial in linking the whole tool to an optimizer to obtain optimized blade shapes with single or multiobjective functions or even an automated process cycle. Design of Experiments approaches can 1

16 also be automated which are often used to create meta models which can be optimized. Also, this design tool provides a bridge between the low fidelity and the high fidelity analysis/design process. Certain CAD features like union of solids and fillets can be achieved by integrating the tool with CAD. 1.2 Literature Review and Previous Work There have been many blade design models developed, which have various approaches for building the blade model. There are several papers on a parametric approach to the blade geometry. One paper by Gräsel et al. [8] describes a complete parametric model for the blade design where the blades are free form shapes defined by Spline-functions. It also describes modification of aerofoils with the help of the curvature distribution defined by control points. Interaction with a CAD interface and how the parameters are optimized is also discussed. Dutta et al. [9] describes an automated process of a non-dimensional quasi-3d blade design with parameterization of the camber-line angle and thickness distributions and blade inlet and outlet angles. It discusses an optimization loop including geometry generation, certain bladeto-blade computations and post-processing, and how each parametric variable has an impact on the optimizer. Kioni et al. [10] describes a tool for parametric design of the turbomachinery blades. Using the design parameters, 2D blade sections are created and using NURBS surfaces, 3D bladings are obtained. These geometries can be exported to any CAD or mesh generation and analysis software. The tool can also be used for design optimization purposes. Various geometries can be handled smoothly due to the object oriented structure of the code. The blade sections are formed by a mean camber line curve and a distribution of control points or interpolation points of a NURBS curve defines the thickness distribution around the mean curve. The desired number of blade sections are constructed and intially, all their leading edges are made to coincide with the origin of the 2D reference system used for the construction. The stacking of the sections is controlled either by a guide line which is a straight line (common radius) or a 3D NURBS curve that passes through the leading edges or through the centres of gravity of the blade sections as explained in the paper [10]. The meridional profiles are generated with the help of the 2

17 control points. Conformal mapping of the 2D blade sections to the cylindrical surfaces using the corresponding meridional curves with respect to a stacking line results in the 3D blade shape. Coordinate transformation from the cylindrical system to the 3D Cartesian system is helpful in constructing the 3D blade surface as a single NURBS surface, where all the sections have the same number of points. The tool also has the capability of creating the hub and the shroud surfaces as a single NURBS surface with the help of the control points. The single curve construction approach minimizes the curvature discontinuity errors and the 3D blade generation is simplified as explained in the paper. Korakianitis [11] describes a method for generating 2D blade shapes. A mixture of analytic polynomials and a desired curvature distribution on a 2D plane is mapped with the blade geometry such that the blade surfaces have continuous slope of curvature throughout their length. The leading edge is defined by 2 thickness distributions around 2 independent construction lines which avoids the overspeed regions near the leading edge. The design method presented in this paper ensures that the boundary layer is not perturbed unintentionally. Effects of stagger angle on blade loading and wake thickness has also been explored. Anders et al. [12] described the blade profile using two Bezier curves of order five. The leading and trailing edges were defined using circles or ellipses. Slope continuity was maintained, except for some curvature discontinuities at certain junctions. The design system consisted of automatic blading, which utilized the parametric approach to create blades for existing and new compressors such as the BR715 Booster, HP9 research compressor, Trent500 and Trent800 HP compressor. The next step was optimizing the 2D blade profiles based on the desired aerodynamic properties. A parametric 3D blade manipulation was carried out by 3D blade stacking followed by radial blade smoothing and interpolation, which created a smooth 3D blade. This whole process was automated and optimized to generate blade shapes satisfying the required objectives. Bezier or B-spline curves are used for airfoil design either as a single closed (or open) curve or as separate curves and introducing circles (or ellipses) at the leading and trailing edge have become popular. Usually, Bezier curves of order 3 or higher are used for the mean camber definition to avoid curvature discontinuites and NURBS are used for joining curves since it 3

18 is the most widely used set of spline curves in many CAD systems. Using control points as parameters proves to be more useful in obtaining desired airfoil shapes. This system is helpful for airfoil design optimization purposes. Several academic or commercial software packages are present for blade design and optimization purposes. BladeCAD, an interactive geometric design tool was developed by Oliver et al. [13] in 1996 under NASA supervision. Using this interface, blade sections were created with respect to general surfaces of revolution and the entire design was defined using a non-uniform rational B-spline (NURBS) surface. The IGES file format was used as the output file which is portable to most analysis and design applications. The blade design was defined by constructing the mean camber line, which is a bezier curve of third degree defined by blade angles and the stagger angle. The shape of the mean camber was specified in the angle preserving space m θ. The blade section was constructed by defining the suction side and the pressure side curves along with the thickness definition normalized by the chord. Using an inverse mapping procedure as explained in the paper, the blade section constructed in m rθ space is converted to its respective stream surface in r z θ space. A stacking axis was defined to loft the blade sections in 3D space. The software has an input panel which contains a thickness function editor, section editor, which uses NURBS curve to define the mean camber geometry, and a blade metal angle editor. It also has the capability of creating general stream surfaces which improves the exisiting blade design methodology. 1.3 Goals and Thesis Layout The work presented in this thesis has been mostly funded by a NASA NRA grant [14], and hence will attempt to meet the goals set forth in that proposal. The unique aspects of this work are: 1. An adherence to consistency. 2. Simplicity of the input while allowing the largest design space. 3. General capability and flexibility of the design tool. 4

19 4. Availability of the code in executable form. Additional goals of this research and the layout of the thesis are: 1. Parameterization of blade geometry inputs which is a very key aspect as it can be modified very easily and can be incorporated in automation and optimization chains. This goal is achieved and is documented in Chapter The development of a stand-alone 3D blade geometry builder called 3DBGB (3 Dimensional Blade Geometry Builder) which produces blade shapes that can be modified to get newer blade shapes by simply updating the spline data with limited interaction with CAD systems. This goal is achieved and is documented in Chapter The integration of 3DBGB into a large, system wide multi-disciplinary analysis and optimization (MDAO) project. Chapters 3 and 4 document progress related to this project. 4. The application of 3DBGB in creating bladeshapes of the 10 stage High Pressure Compressor developed from the GE/NASA program [1], both free vortex and full definition design. This is documented in Chapter The application of 3DBGB in creating bladeshapes of a 3-stage booster for a turbofan engine and performing a CFD and FEA analysis on rotor 3. This is documented in Chapter A reverse engineered wind turbine design was also created using this tool to demonstrate the generality of the design tool and is documented in Chapter The development of a general blade geometry builder which creates both axial and radial turbomachinery blades. This is demonstrated by designing a radial compressor as explained in Chapter 5. 5

20 Chapter 2 3D Blade Generator This chapter deals with the development of the blade generator explaining how the input parameters are utilized in creating blade sections and performing mathematical operations to finally obtain a 3D blade model. The 3DBGB [15] code creates the 3D blade shapes presented as examples. The executable with several cases is available at Figure 2.1 explains the process flow involved. 2.1 Input File The input file for the 3DBGB (3 Dimensional Blade Geometry Builder) code [15] contains aerodynamic and geometric parameters defining the blade geometry required to build a 3D blade shape. The parameters are defined at each streamline passing through the blade as below: 1. flow angles at leading edge and trailing edge of the blade. 2. relative inlet mach number. 3. thickness to chord ratio. 4. axial chord values. 5. incidence and deviation angles (if these are zero, then the flow angles represent the blade metal angle). 6

21 Figure 2.1: 3D Blade Design Process Flowchart. 7

22 The input also has 2D curves representing the leading and trailing edges, airfoil stacking information and a control table which defines the sweep and lean perturbation and any required quantity with a few control points for adding more definition to the blade geometry. The input file also contains the x s and r s coordinates for all the nspn streamlines defined from an axisymmetric run like T-Axi [16, 17, 18, 19], or by smooth construction curves between the hub and the casing. 2.2 Governing Equations Figure 2.2: r x θ space for the 3D blade. The coordinate system used is shown in Figure 2.2. The meridional view of the streamlines with the leading edge and the trailing edge is shown in Figure 2.3. The curve in Figure 2.3 is referred to as a streamline since an axisymmetric streamline has traditionally been used as a construction line or curve. A smooth construction curve can be used in place of an axisymmetric streamline to define the blade geometry. The 3D blade is constructed using the following mathematical approach: 1. streamline coordinates: m s, x s, r s. 2. airfoil coordinates: m b, θ b. The projection of the streamline or smooth construction curve onto the meridional plane 8

23 Figure 2.3: Meridional view of the blade showing axisymmetric streamlines as construction curves. x r is given by: dm s = (dr s ) 2 + (dx s ) 2 (2.1) The normalized differential arc length is defined by: dm s = dm s r s (2.2) The m s coordinate of the streamline is obtained by integrating: m s = dms (drs ) = 2 + (dx s ) 2 (2.3) r s r s If the airfoil is designed on constant radius sections, then m dxs s = r s = x s r s (2.4) which represents a normalized axial coordinate. 9

24 2.3 Blade section construction Figure 2.4: A blade showing the blade sections. The blade section construction procedure documented in this section is adopted from the procedure used in T-Axi [16, 17, 18, 19] co-authored by Dr. Ali Merchant. The same procedure is also used in T-Axi Blade [20] which is a GUI based, free-vortex blade row design and visualization program. It s main emphasis is on the relationship between velocity triangles and blade shapes, cascade views for hub, pitch and tip, Smith charts and Loading vs Aspect Ratio charts. A 3D blade contains a specified number of blade sections stacked radially as shown in Figure 2.4. A blade section contains a mean camber curve passing through the leading edge and trailing edge of the blade section and has a suction side curve and a pressure side curve built around it as shown in Figure 2.5. Also in Figure 2.5, the blade metal angle at inlet (leading edge) is defined by βin and at the exit (trailing edge) is defined by βout. The flow angles at inlet and exit are β in and β out respectively. In this figure, βin, βout and ζ are all negative based on the 10

25 sign convention used. The incidence angle i and the deviation angle δ are given as below: 1. Incidence Angle, i: If β in greater than zero then i = β in β in and if β in is less than zero then i = β in β in. 2. Deviation Angle, δ: If camber is negative then δ = β out β out and if camber is positive then δ = β out β out. Figure 2.5: Blade section on the mean camber curve. An elliptical leading edge and circular trailing edge are used for connecting the pressure side and suction side curves. The elliptical leading edge helps in reducing overspeeds and for certain Reynolds numbers, helps to keep the flow attached and laminar [21]. Also, the user has an option of choosing the shape of the leading and trailing edges. The mean camber line is built using the blade metal angles (βin, βout), thickness to chord ratio and the meridional chord value. A mixed camber line is defined which is partly cubic in nature. The analytical form of the camber line cam is a cubic polynomial which is a function of a non-dimensional coordinate u. v is the corresponding non-dimensional coordinate perpendicular to u. The relationship between (m b, θ b) system and (u, v) system is shown in the Figure

26 Figure 2.6: Relationship between (m b, θ b) system and (u, v) system. The analytical form of the camber line cam is: cam = aa(ub) 3 + bb(ub) 2 + cc(ub) + dd (2.5) cam u = 3aa(ub) 2 + 2bb(ub) + cc (2.6) ub = u u 1 (2.7) xb = u 2 u 1 (2.8) aa = s 1 + s 2 2( c 2 dd xb ) (xb) 2 (2.9) bb = s 1(xb) aa(xb) 3 + c 2 dd (xb) 2 (2.10) cc = s 1 (2.11) dd = c 1 (2.12) In Eq.(2.5), aa, bb, cc and dd are the coefficients of the cubic equation representing the 12

27 camber line and ub is the varying parameter. Also, s 1 = tan βin (2.13) s 2 = tan βout (2.14) u 1 = fl 1 cos βin (2.15) c 1 = fl 1 sin βin (2.16) u 2 = 1 fl 2 cos βout (2.17) c 2 = fl 2 sin βout (2.18) where, 1. s 1, s 2 are the slopes at inlet and exit of the blade section. 2. fl 1 = constant slope at inlet as seen in Figure fl 2 = constant slope at exit as seen in Figure c 1, c 2 are the derivatives of u 1, u 2 with respect to the blade metal angle. The camber line plot of the 10th and 11th blade sections for rotor 1 blade of the 10 stage high pressure compressor developed from GE/NASA program [1] is shown in Figure 2.7. Figure 2.8 shows the plot of camber angle for the same rotor blade sections. Also, the curvature of the camber line for the 10th and 11th blade sections are plotted in Figure 2.9. All these plots are with respect to u, the non-dimensional coordinate spanning the interval [0,1]. An equally spaced grid of 100 points (can be varied) in this interval was used for the plot. The camber angle, ang is given by differentiating Eq. (2.5) with respect to ub and taking the arctangent of the value obtained. ang = tan 1 (cam u ) (2.19) The non dimensional coordinates for suction side (u btop, v btop ) and pressure side (u bbot, v bbot ) are 13

28 Figure 2.7: Plot of the camber line of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design. calculated as below: u bbot = u + thk sin (ang) (2.20) v bbot = cam thk cos (ang) (2.21) u btop = u thk sin (ang) (2.22) v btop = cam + thk cos (ang) (2.23) where thk is the thickness distribution defined over a non-dimensional unit camber length (0.0 to 1.0) as given by Wennerstrom [2]. The thickness distribution is described in Figure 2.10 and the nomenclature is as below: 1. x = non-dimensional camber-line length. 2. x T = location of maximum airfoil thickness. 3. t 1 = leading edge thickness. 4. T = maximum airfoil thickness. 5. t 2 = trailing edge thickness. 6. y 1 = airfoil half-thickness upstream of x T. 14

29 Figure 2.8: Camber angle plot of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design. 7. y 2 = airfoil half-thickness downstream of x T. The thickness equation for x < x T : y 1 = ax 3 + bx 2 + cx + d (2.24) y 1 = 3ax 2 + 2bx + c (2.25) y 1 = 6ax + 2b (2.26) 15

30 Figure 2.9: Plot showing the curvature of the camber line of blade sections 10 and 11 of the rotor 1 blade from the 10 stage HPC design. Figure 2.10: Thickness distribution over a non-dimensional unit camber length [2]. and the thickness equation for x > x T : y 2 = e(x x T ) 3 + f(x x T ) 2 + g(x x T ) + h (2.27) y 2 = 3e(x x T ) 2 + 2f(x x T ) + g (2.28) y 2 = 6e(x x T ) + 2f (2.29) When x T 0.5, the boundary conditions are: 16

31 At x = 0 y 1 = t 1 2 (2.30) y 1 = 0 (2.31) At x = x T y 1 = y 2 = T 2 (2.32) y 1 = y 2 = 0 (2.33) y 1 = y 2 (2.34) At x = 1.0 y 2 = t 2 2 (2.35) (2.36) From these, the coefficients a through h are derived as explained by the author, Wennerstrom in his book [2]. When x T < 0.5, the leading and trailing edge boundary conditions are reversed to obtain the coefficients. The suction side and pressure side coordinates are merged into a single array of specified number (199 in this case) of non-dimensional coordinates (u b, v b ) respectively, such that the array starts from the trailing edge and goes counter-clockwise through the leading edge and comes back to the trailing edge. Since the leading edge and trailing edge coordinates of the pressure side and suction side are the same, only a single value of the leading edge coordinate is used in the array and the first and last point in the array is the same trailing edge coordinate. The non dimensional leading edge is chosen as (0,0) and trailing edge as (1,0) for convenience. The stagger angle, ζ is calculated by taking the average of the blade metal angles at inlet and 17

32 exit. The normalized staggered airfoil coordinates (m bstgr, θ bstgr) are as given below: ζ = (β in + β out)/2 (2.37) m bstgr = u b cos ( ζ) + v b sin ( ζ) (2.38) θ bstgr = u b sin ( ζ) + v b cos ( ζ) (2.39) Figure 2.11 shows the airfoil rotated by the stagger angle and is placed on the unit chord on the non-dimensioanl spacing array system (u, v). chrdx is the non dimensional meridional chord which is the difference between m T E and m LE. The actual chord is calculated and the airfoil coordinates are scaled by multiplying the staggered coordinates (m bstgr, θ bstgr) with the non-dimensional actual chord as below: chrdx = m T E m LE (2.40) chrd = chrdx/ cos ζ (2.41) m b = m bstgr(chrd) (2.42) θ b = θ bstgr (chrd) (2.43) Figure 2.11: Blade section on the normalized unit chord. The resultant 2D blade section contains np values of m b, θ b coordinates in the meridional coordinate system. 18

33 2.3.1 Variety of Airfoil shapes 3DBGB is flexibile in its capability of generating different airfoil shapes to cope with a variety of blade designs. Since each blade section is constructed independently through parameters, it is possible to have different blade sections at different spanwise locations. For example, a wind turbine blade has a circular hub and progresses towards an S809 airfoil shape radially. The general capability of the blade section construction makes it easy to deal with such shapes. Currently, 3DBGB is capable of generating the S809 airfoil, NACA 4-digit airfoils, a circular section, and the default airfoil which is a mixed camber airfoil as shown in Figure In addition to these, the tool is flexible in adding various airfoil shapes parametrically. It is recognized that the default blade section is inadequate for a transonic fan, but those sections can be added in a parametric fashion. This is also true for many other blade types. The integrated ability for a general blade section and consistent 3D stacking in a rigorous manner sets this blade generator apart. Figure 2.12: Variety of airfoils generated by 3DBGB in addition to circular and 4 digit NACA airfoils. 2.4 Leading Edge and Trailing Edge Curves The leading and trailing edge are defined by 2D cubic spline curves in x, r. A cubic spline is used to fit the streamline coordinates x s, r s with m s as the spline parameter. The intersection of these curves and the 2D streamline curves produces the leading and trailing edge coordinates 19

34 (x LE, r LE, x T E, r T E ) on the corresponding streamlines. This information is used to obtain the m sle, the leading edge m s value, which is crucial in streamwise mapping of the blade sections. 2.5 Streamwise Mapping of the Blade Sections The 2D blade sections created are mapped with their corresponding streamlines. The streamlines are defined both upstream and the downstream of the blade for robust mapping. Typical streamsurfaces are defined by revolving the streamlines about the x-axis. For an axial and radial machine this is shown in Figure m sle, the leading edge m s value is calculated by taking the inverse spline of x(m s) evaluated at x LE on each of the streamline as shown in Figure The meridional offset between the blade leading edge m ble and the streamline leading edge Figure 2.13: Stream surface for an axial machine and a radial machine respectively. Figure 2.14: m sle obtained using inverse spline on m s. 20

35 m sle is calculated and is necessary for precise conformation of the blade section on the corresponding streamline. This is because the zero m s on each streamline is different from the m ble on each corresponding blade section and this offset is expressed as: δm = m sle m ble (2.44) Similarly, there exists a tangential offset which is added to the θ b coordinates to obtain streamwise θ 3D coordinates for the blade: θ 3D = θ b + δθ (2.45) Once all the offsets are calculated, the streamwise meridional coordinates m 3D for each blade section is obtained using: m 3D = m b + δm (2.46) The m 3D coordinates are used to calculate the streamwise x 3D and r 3D coordinates by evaluating the spline at every streamwise meridional coordinate as below: 1. x 3D = spline evaluated at each m 3D for all the np points of each blade section. 2. r 3D = spline evaluated at each m 3D for all the np points of each blade section. The spline evaluation process is depicted in Figure 2.15 and Figure Figure 2.15: x 3D obtained by spline evaluation at each m 3D of each blade section. Figure 2.17 shows the mapping of the blade section on the corresponding stream surface. In this manner, all the blade sections are mapped consistently. x 3D, r 3D and θ 3D coordinates for 21

36 Figure 2.16: r 3D obtained by spline evaluation at each m 3D of each blade section. Figure 2.17: Mapping of the blade section to the corresponding stream surface. all the nspn 3D blade sections are thus calculated in the cylindrical coordinate system as shown in section Sweep and Lean to the Blade Spanwise Sweep and Lean are applied as perturbations using a specific number of control points (span, δm ) and (span, δθ). The control points are used to construct a smooth sweep and lean definition spanwise using uniform B-splines as explained later in this section. The normalized 22

37 span of the n th streamline for an axial machine is given by : L ref = r LET IP r LEHUB (2.47) span(n) = r(n) r HUB (2.48) span(n) = span(n)/(l ref ) (2.49) where L ref is the blade height. The sweep perturbation, δm values at each streamline are evaluated by interpolating the δm curve at each spanwise location on each streamline as depicted in Figure In this manner, the sweep is added to the blade whenever required. Figure 2.19 shows blades with forward and backward sweep respectively. Figure 2.18: δm evaluated at each span. Similarly, the lean is added to the blade as required by defining the lean perturbation with few control points and obtaining the δθ values at each streamline as explained above. This method as shown in Figure 2.20 was used to modify the lean near the hub of rotor 3 of the booster optimum shown as point 2 in the paper by Park et al. [4] to eliminate the corner separation at the exit which increased the adiabatic efficiency for the isolated blade row 3D CFD analysis of rotor 3 as explained by Siddappaji et al. [15]. 23

38 Figure 2.19: Forward and backward swept blades respectively. Figure 2.20: Spanwise lean modification on rotor 3 of a booster optimum resulting in lean near the hub Construction of Uniform B-Splines A group of local control points are used to determine the geometry of curve segments which form a B-spline. A curve segment does not necessarily have to pass through a control point but its desired at the two end points of the B-spline. Cubic B-splines are popular because of their continuity characteristics which make the segment joints smooth [3]. Figure 2.21 shows a cubic curve constructed from a series of curve segments S 0, S 1, S 2,..., S m 3,... using m + 1 control points P 0, P 1, P 2,..., P m. Any single segment S i (t) of a B-spline curve is defined by S i (t) = 3 P i+r B r (t) (2.50) r=0 24

39 Figure 2.21: Construction of a uniform non-rational B-spline curve. [3] where t (0 t 1) is a parameter and the B-spline basis functions are as below [3] B 0 (t) = t3 + 3t 2 3t B 1 (t) = 3t3 6t B 2 (t) = 3t3 + 3t 2 + 3t B 3 (t) = t3 6 = (1 t)3 6 (2.51) (2.52) (2.53) (2.54) The B-spline curve, Q 1 (t) is represented in matrix form [3] by [ ] Q 1 (t) = t 3 t 2 t P i P i+1 P i+2 P i+3 (2.55) where P i, P i+1, P i+2 and P i+3 are the 4 consecutive control points of the total number of control points. So, if there are 5 control points then the first curve segment is made up of the first 4 control points (1, 2, 3, 4) and the second curve segment is made up of the second 4 consecutive control points (2, 3, 4, 5). This method of construction is used in adding the lean and sweep to the blade. Once the B- 25

40 spline is constructed, a simple 2D-curve and a line intersection procedure will give the desired point on the spline curve. For example, spanwise sweep is defined by the control points (span, δm ). A B-spline curve of δm is constructed as explained above. The δm value at a particular spanwise location is obtained by intersecting the 2D δm curve with the line (parallel to X-axis) passing through that location as shown in Figure Radial Stacking of 3D Blade Sections The 3D blade sections are stacked radially by including a θ offset, θ stack for each individual blade section. Stacking is defined as a fraction of the non-dimensional actual chord. This value is transformed into the average θ 3D coordinate, which is subtracted from the array of θ 3D coordinates to obtain the stacking. The stacking can be done at the leading edge, trailing edge, at any percentage of the chord and at the area centroid of a blade section. θ 3D = θ b + δθ + θ stack (2.56) Figure 2.22 explains the theta offset defined for stacking the blade at the leading edge, percentage chord and the trailing edge correspondingly. Finally, using the coordinate transformation the 3D blade coordinates in the cartesian system is obtained for this case. 2.8 Coordinate Transformation The normal practice is to obtain a 3D blade in the cartesian coordinate system as most of the CAD packages exist in this system. Therefore a coordinate transformation from the cylindrical system to the cartesian system is necessary. The engine axis is assumed to be along the X-axis which makes the x-values remain the same in both coordinate systems. The transformation is as follows: 26

41 Figure 2.22: θ offset defined for various stacking option. x 3D = x 3D (2.57) y 3D = r 3D sin θ 3D (2.58) z 3D = r 3D cos θ 3D (2.59) 2.9 Output files The blade generator code outputs nspn data files, which contain np values of 3D coordinates for all the nspn blade sections. The number of coordinates in the 3D blade sections are kept the same as the number of coordinates in the 2D airfoil sections. The 3D blade section files can act as input files for any CAD package to obtain a 3D Blade CAD model. 27

42 2.10 3D Blade CAD Model All the data files are imported in Unigraphics (CAD package) and a lofting procedure is performed to create 3D blade surfaces passing through the blade sections to obtain a smooth 3D blade as shown in the Figure The parameters used in lofting are the 3D spline data obtained as the 3D blade section data from 3DBGB for each blade section. Figure 2.23: 3D Blade lofted in UG Connecting with CAPRI CAPRI stands for Computational Analysis PRogramming Interface. The 3D Blade constructed in UG is a base model and using CAPRI [22, 23], newer blade shapes are obtained by simply updating the spline information of each 3D blade section on the base model. A simple program written in C integrates the 3D blade part file and the CAPRI interface through which the spline update is done and hence the blade geometry is morphed parametrically. The advantage of using CAPRI is that the geometry data remains in the CAD system and avoids the geometry translation errors during morphing of the blade geometry. CAPRI saves significant time and effort in development, deployment, and maintenance of multi-disciplinary design suites that need to interface with CAD. It does not require low-level expertise in CAD or CAD programming to use, and provides an intuitive engineering interface to CAD for MDAO applications. 28

43 2.12 Extruded Blade In some cases, grid generators require an extruded blade due to tolerance issues and extrusion is a property which is useful for other purposes as well. It is achieved through a simple offset of the hub and the tip streamline coordinates in the normal direction to the streamline. At any point m s, the normal in the x-direction, x NORM and the normal in the r-direction, r NORM are calculated by using the orthogonal property between the normal and the slope at that point. x NORM = dr s dm s r NORM = dx s dm s (2.60) (2.61) So, the offset in the normal direction as shown in Figure 2.24 is given by n = al ref, (2.62) where a is the percent offset desired and L ref is the reference length. Figure 2.24: Offset at the hub streamline. Also, n = b x NORM 2 + r NORM 2 (2.63) where b is an intermediate extruded blade scale factor. b is solved using the 4 equations above, and the offset at the hub and the tip is obtained as below: 1. streamline coordinates at the hub : x shub, r shub 29

44 2. streamline coordinates at the tip : x st IP, r st IP x sextruded = x shub + (b x NORM ) (2.64) r sextruded = r shub + (b r NORM ) (2.65) and the offset at the tip as x sextruded = x st IP (b x NORM ) (2.66) r sextruded = r st IP (b r NORM ) (2.67) The new extruded hub and tip streamline coordinates are used instead of the original coordinates and the blade parameters corresponding to the original hub and tip streamlines are used as the input to obtain an extruded 3D blade as shown in Figure 2.25 using the procedure as explained before. Figure 2.25: Extruded blade at the tip with hub and casing. 30

45 Chapter 3 CFD Analysis of the 3D blade A CFD simulation is used to analyze the resulting 3D blade geometry. F INE T M /T urbo v8 by Numeca [24] was used for the 3D CFD analysis on the blade model obtained. F INE T M /T urbo is a high fidelity package which has its own gridding tool, solver and a post processor. 3.1 Gridding The 3D blade section geometry created along with the hub and the shroud definition was imported into a gridding tool called AutoGrid5 [25]. The blade geometry is the rotor 3 blade of the 3 stage booster from a paper by Park et. al [4]. The blade model thus imported is given a tip clearance and other necessary input details such as the rotational speed and type of turbomachinery system. The tip clearance is 0.15% of rotor tip radius which is representative of a compressor with tight but realistic tip clearance. The grid generated is medium type with grid points. The flow path is generated and blade to blade mesh is created as shown in Figure CFD solver: Euranus Euranus, the F INE T M /T urbo solver is run with inlet boundary conditions of absolute total pressure, absolute total temperature, spanwise distribution of α x at inlet and φ coming from the previous blade row and static pressure as the exit boundary condition. An isolated blade row 31

46 Figure 3.1: Creating the meridional and blade to blade mesh. 32

47 3D CFD analysis of rotor 3 of the booster optimum shown as point 1 in paper by Park et. al [4] has been performed. Because of the thick boundary layer of the axisymmetric analysis, the first displaced streamline was used to define the hub and the last displaced streamline was used to define the casing. The simulation resulted in an adiabatic efficiency of 91.25% at the design flow rate. This compares 93.47% from the T-Axi axisymmetric code loss model [17]. Several runs were made on this case by varying the back pressure to create a speed line (the design mass flow rate is kg/s). Figure 3.2 shows the mass flow rate variation with the total pressure ratio and Figure 3.3 shows the variation of mass flow rate with the isentropic efficiency and the wide range of the mass flow rate for the booster can be noticed. Figure 3.4 shows the contour plot of relative mach numbers from 0.2 to 0.8 on three constant radius cut-planes across rotor 3 of the booster. It also shows stream ribbons across the tip clearance showing the tip vortex. Reasonable agreement between the axisymmetric and 3D demonstrates how the coupled system can be used for optimizing a design. Figure 3.2: Characteristic curve showing Total PR vs mass flow rate. The isolated blade row 3D CFD analysis of rotor 3 of the booster optimum shown as point 2 in the paper by Park et. al [4] was also performed which resulted in corner separation at the 33

48 Figure 3.3: Characterisitic curve showing Isentropic efficiency vs mass flow rate. Figure 3.4: Relative Mach contours of rotor 3 of the optimized booster with stream ribbons at over the tip clearance. 34

49 exit with an adiabatic efficiency of 93.17%. The lean of the 3D blade geometry was modified as shown in Figure 3.5 which eliminated the separation near the exit as shown in Figure 3.6 which is the comparison of the iso surface of axial velocity for the original and modified rotor 3 for this case. The adiabatic efficiency increased to 93.93%. This compares to the T-Axi efficiency of 95.27% for this rotor. It should be noted that this is still not an optimum. The shape shows that there might be stress issues except that the wheel speed is so low which is why the coupling with a finite element structural solver is so important. Figure 3.5: Spanwise lean modification on rotor 3. Figure 3.6: Iso surface comparison of axial velocity for original and modified rotor 3 of the booster optimum point 2 [4]. 35

50 Chapter 4 Structural Analysis of the 3D Blade Structural Analysis is a critical design simulation capability and its use is demonstrated on the blade geometry. ANSYS V12.0 [26] is used. A CAD part file of the 3D blade model is imported and a script file called Blade.ain is used which contains the material properties of the blade and also the instructions for meshing the blade. This creates a meshed blade with hexagonal mesh as shown in Figure 4.1 and is ready for structural analysis. The blade model used is the rotor 1 blade of the 10 stage High Pressure Compressor. A modal analysis is performed with 14 modes and the procedure is as follows: Analysis type is defined. Solution Analysis Type New Analysis Modal. Analysis option is set. Solution Analysis Type Analysis Options Preconditioned Conjugate Gradient (PCG) Lanczos. Number of modes to extract is 14 and it extracts modes for all Degrees Of Freedoms (DOF s). Constraints are applied. Solution Define Loads Apply Structural Displacement On Areas. The hub area of the blade is selected. 36

51 Figure 4.1: Blade meshed in ANSYS V12.0. The system is solved. Solution Solve Current Load Step (LS). After the solution is complete post processing is performed. Figures 4.2 and 4.3 shows the contour plots of the Displacement vector sum of Degrees Of Freedom (DOF) solution under Nodal solution for all the 14 modes of the blade. The plots are of a deformed shape of the blade. Figure 4.2: Contour plots of Displacement vector sum for first 7 modes of a blade under Nodal solution. 37

52 Figure 4.3: Contour plots of Displacement vector sum for last 7 modes of a blade under Nodal solution. Also, Figure 4.4 shows the displaced blade with the undisplaced blade shape keeping the hub of the blade fixed. This completes the structural analysis of the blade. Figure 4.4: Displaced blade. 38

53 Chapter 5 Example Design Problems The capability of the 3DBGB is demonstrated through the following examples. The examples include a 10 stage EEE high pressure compressor design, a 3 stage booster, a low speed centrifugal compressor, all of which are generated using the design data from reports. A wind turbine design is also demonstrated, and has been reverse engineered from pictures, marketing literature [6] and a report [27]. All these example cases are available at Stage EEE High Pressure Compressor Design The GE Energy Efficient Engine (E 3 ) high pressure compressor (HPC) as shown in Figure 5.1 was used as an example, which has 21 blade rows with 10 stages and an IGV. Two cases were generated, one with free vortex having straight leading and trailing edges and the other with full definition of spanwise angular momentum having non-straight leading and trailing edges Free vortex Design The angular momentum is kept constant along the blade span for all the blade rows for this case and it has straight leading and trailing edges. The input files for running T-Axi (axisymmetric solver) [16, 17, 18, 19] for this case are available on the T-Axi website[16] under input.zip/t-axi Input Files/EEE-HPC-des. The axisymmetric grid of this 10 stage HPC (free vortex) is shown in Figure 5.2. The grid is created by the hub and casing definitions used in 39

54 Figure 5.1: E 3 10 Stage Compressor Cross Section. [1] the solver and the green lines represent the leading and trailing edges of the blade rows coming from the T-Axi input file. The T-Axi solution provides information about flow angles, relative mach number and other Figure 5.2: Axisymmetric grid showing the streamlines of 10 Stage HPC free vortex design and the straight leading and trailing edge locations of the blade rows in green. aerodynamic data at each streamline along with providing the location of the leading and trailing edges on each streamline. It also outputs the input file for 3DBGB, containing the required aerodynamic and geometric parameters to generate the 3D blade shape. For this case, the flow angles are used as the blade metal angles to generate the airfoil sections. All the values of coordinates were normalized using the leading edge tip radius of rotor 1. The chord is internally calculated. The maximum thickness to chord ratio for rotors is defined linearly from 0.1 at the hub to at the tip, for stators from 0.05 at the hub to 0.1 at the tip and for IGV a constant value of These numbers are based on the GE EEE report [1]. The 3D blade shape has 10 blade sections. One of the 3D blade is built using 3DBGB and lofted using UG-NX. The rest of 40

55 the blades are obtained by updating the spline data on the initial blade using CAPRI [22, 23]. The spline data is produced as 3D blade section data for the respective blades by 3DBGB. The blades are stacked at their leading edge for this case. The rotor 1 and rotor 3 blade shapes for the free vortex design are shown in Figure 5.3. Figure 5.3: Rotor 1 and rotor 3 of EEE HPC free vortex design created using 3DBGB. In this manner, all the 21 blade rows are constructed and are assembled with their corresponding disks optimized with T-Axi Disk [5, 20] to obtain blisks. Rotor 1 and rotor 3 blisks are shown in Figure

56 Figure 5.4: Rotor 1 and Rotor 3 blisk assembly for EEE HPC free vortex design. 42

57 5.1.2 Full Definition 10 stage Design The angular momentum along the blade span is not constant for the E 3 case and it has nonstraight leading and trailing edges. The input files for running T-Axi for this case are available on the T-Axi website [16] under input.zip/t-axi Input Files/EEE-HPC-analyse. The axisymmetric grid of the GE EEE HPC (full definition) is shown in Figure 5.5. The grid is created by the hub and casing definitions used in the solver and the green lines represent the leading and trailing edges of the blade rows coming from the T-Axi input file. Figure 5.5: Axisymmetric grid showing the streamlines of E 3 10 Stage HPC full definition design and the leading and trailing edge locations of the blade rows in green. The T-Axi solution gives flow angles and other aerodynamic data for all the specified 21 streamlines and also outputs 3DBGB input containing required information for 21 streamlines. For this case, the blade metal angles from the NASA report[1] were used to construct the airfoil sections. The NASA report contains data tables of blade metal angles, maximum thickness to chord ratio and other parameters tabulated spanwise for all 21 bladerows of the GE EEE High Pressure Compressor. The report has data tabulated at 12 spanwise locations for each blade. These 12 values of the blade metal angles and the maximum thickness to chord ratio were interpolated to 21 values since this 3DBGB input has 21 spanwise locations. All the values of coordinates were normalized using the leading edge tip radius of rotor 1. A spanwise comparison of the 12 values of blade metal angles (β1, β2) at inlet and exit of rotor 3 as given in the NASA report [1] with 21 interpolated values obtained by 3DBGB is shown in Figure 5.6. Using these interpolated values and the internally calculated leading and trailing edge locations, a 3D blade shape with non-straight leading and trailing edge is constructed for rotor 3 in this case and is shown in Figure 5.7. The first 5 rotors were stacked at the center of the 2D airfoil 43

58 Figure 5.6: Comparison of blade metal angles of EEE-HPC rotor 3 given in the report [1] and generated by 3DBGB. area and the rest were stacked at quarter chord radially. All the other 20 blades are constructed using CAPRI and respective spline data as explained before. A complete assembly with all the 10 rotors, 10 stators and the IGV is created in UG without the hub and the casing definition and is shown in Figure 5.8. Figure 5.7: Rotor 3 of GE EEE HPC full definition design with non-straight leading and trailing edges. 44

59 Figure 5.8: Rotor and stator assembly for 10 stage HPC full definition design from the GE/E 3 parameters. 45

60 5.2 3 Stage Booster Design for Turbofan Engine A 3 stage booster for a turbofan engine (CFM56 class machine) was designed. Boosters are used in turbofan engines before the high pressure compressor. The 3 stage booster of the GE90 turbofan engine shown in Figure 5.9 depicts where the boosters are used. Figure 5.9: 3 stage booster of a GE90 turbofan engine.[5] A 3 stage booster flow path geometry along with the blades was optimized as explained in the paper by Park et. al [4] and the best result was used for generating 3D blades. The axisymmetric grid of the optimized booster is shown in Figure 5.10 which shows the streamlines calculated by T-Axi and the leading and trailing edges of blade rows in green lines. The axisymmetric solver (T-Axi) is run and the input for 3DBGB was obtained. The flow angles are used as the blade metal angles for the 2D airfoils construction. The maximum thickness to chord ratio for rotors is defined linearly from 0.1 at the hub to at the tip and for stators from 0.05 at the hub to 0.1 at the tip. All the blade rows are created as explained in the previous example and the rotors and stators are assembled as shown in Figures 5.11 and The validation of rotor 3 was done by performing a 3D-CFD analysis which is explained in Chapter 3. The FEA structural analysis was also performed on rotor 3 with a tetrahedral mesh as shown in Figure A modal analysis on the blade resulted in 14 modal shapes and the 46

61 Figure 5.10: Axisymmetric grid of the 3-stage optimized booster showing the streamlines and the leading and trailing edge locations of the blade rows in green. Figure 5.11: 3 stage booster rotor assembly with the fan blades. 47

62 Figure 5.12: 3 stage booster assembly with split casing. 48

63 contour plots of the displacement vector sum is shown in Figure Figure 5.13: Tetrahedral mesh of rotor 3 of a 3 stage booster in ANSYS. Figure 5.14: Contour plots of displacement vector sum of all 14 modes of rotor 3 under Modal solution. 49

64 5.3 Reverse-Engineered Wind Turbine Design Wind turbines can be treated as turbines without the nozzle. All the turbomachinery principles apply to them. 3DBGB is capable of creating wind turbine blades due to its generality. This capability is demonstrated by building the reverse engineered GE 1.5 MW wind turbine blade as shown in Figure Pictures, marketing literature [6] and a report [27] were used for reverse engineering. Figure 5.15: GE 1.5 MW wind turbine blade. [6] The case for T-Axi was set up as explained in the Masters thesis by Dey [28]. The axisymmetric grid of the reverse engineered GE wind turbine blade is shown in Figure The wind turbine was treated as a half stage turbine with extended tip in T-Axi solver. T-Axi generates the input for 3DBGB and the wind turbine 3D blade geometry is created as shown in Figure It is then assembled with the bullet nose and a wind turbine model with 3 blades is created using UniGraphics as shown in Figure While generating a similar wind turbine blade shape, one of the issues encountered was going beyond axial (ζ equal to 92 degrees), where the blade shapes failed to be generated since the chrd value calculated according to the Eq.(2.41) became a non-defined value as cosine of the stagger angle reaches zero. To overcome this issue, a switch was added in the input file which would read the actual non-dimensional chord values chrd for such high staggered cases instead of using Eq.(2.41). The addition of this option expands the capability of the tool to include blades with very high stagger. The utility of the blade generation is demonstrated by performing CFD analysis as explained in the Masters thesis by Dey [28]. FEA structural analysis was also performed by treating the blade as a rotating cantilever beam. The meshed blade has 8 node hexahedron brick 185 element 50

65 Figure 5.16: Axisymmetric grid of the reverse engineered GE 1.5 MW wind turbine blade. Figure 5.17: Reverse engineered GE 1.5 MW wind turbine blade geometry created using 3DBGB. 51

66 Figure 5.18: Reverse Engineered GE 1.5 MW wind turbine assembly. 52

67 as shown in Figure Structural analysis was executed and the Von-Mises plot is shown in Figure Aluminium material was used for the blade which has the yield strength of 414 MPa. The maximum value of the von-mises stress obtained was MPa, which is less than one third times the yield strength. Figure 5.19: Meshed wind-turbine blade with 8 node Hexahedron Brick 185 element. Figure 5.20: Von-Mises Stress plot on GE reverse engineered blade. The first 5 mode shapes for the reverse engineered GE wind turbine blade is shown in Figure

68 (a) Mode Hz (flapwise). (b) Mode Hz (edgewise). (c) Mode Hz (flapwise). (d) Mode Hz (mixed). (e) Mode Hz (mixed). Figure 5.21: First 5 Modal solution of GE reverse engineered blade. 54

69 5.4 Low-Speed Centrifugal Compressor design 3DBGB is also capable of designing 3D blades for radial machines. The low-speed centrifugal compressor (LSCC) blade model is generated as an example case. The compressor is based on a NASA design [7] and has 20 blades with the speed of 1862 rpm. The flowpath of the compressor with the actual experimental setup measurements is shown in Figure Figure 5.23 shows the radial grid of the compressor with hub and casing definition and the streamlines along with leading edge and trailing edge locations denoted by green lines. Figure 5.22: LSCC flowpath with actual measurements. [7] The NASA report [7] contains hub and tip streamline coordinates and other airfoil geometry data. The hub and tip flowpath were linearly interpolated in the radial direction to generate more construction lines. A total number of 5 construction lines including the hub and tip streamlines were generated. All the coordinates were normalized with the leading edge tip radius value. The inlet was treated to be purely axial with constant axial velocity, V z. The wheel speed at the tip, U t is given to be 153 m/s. The report had spanwise plots of Vz U t ratios and an average value of 0.4 at 50% pitch was used to calculate the constant axial velocity as 61.2 m/s. Figure 5.24 shows the velocity triangle for the purely axial flow at the leading edge, where the relative flow angles at the leading edge for different spans are calculated as below: 55

70 Figure 5.23: A radial compressor flowpath showing the hub and casing with streamlines and the leading and trailing edge locations. 1. V z = Axial flow velocity. 2. U = Wheel speed. 3. N = Wheel rotational speed in RPM. 4. W = Relative flow velocity = U = ωr. 5. r = radius in meters. 6. ω = 2πN W θ = Tangential relative flow velocity. 8. β z = Relative flow angle = tan 1 W θ V z. The outlet was treated to be purely radial with constant relative flow angle of 55 o spanwise. The design methodology used to create axial blades was applied except, the blade was stacked at the trailing edge. Also, a condition was added to check the slope of the streamline to distinguish between radial and axial machines. When, dr s > dx s (5.1) dm s dm s 56

71 Figure 5.24: Velocity triangle at the leading edge for the radial compressor. normalized r s values (in radial cases) are used to obtain m T E (or m LE when the flowpath starts as radial) through inverse spline, instead of the usual method of using normalized x s values (in axial cases) while calculating the non-dimensional meridional chord. The 3D blade sections are created and stacked with the trailing edge as the stacking axis. Once the blade sections are obtained, they are lofted using Unigraphics-NX to obtain a solid blade model. The assembled compressor with 20 blades and its comparison to the model from the NASA report [7] is shown in figure Figure 5.25: Radial compressor model compared with NASA LSCC [7]. 57