COMPUTATIONAL VALIDATION OF THE COMPRESSOR DESIGN PROGRAM BLADE LAYOUT METHOD MATEJ ZNIDARČIĆ. Submitted in partial fulfillment of the requirements

Transcription

1 COMPUTATIONAL VALIDATION OF THE COMPRESSOR DESIGN PROGRAM BLADE LAYOUT METHOD by MATEJ ZNIDARČIĆ Submitted in partial fulfillment of the requirements For the degree of Master of Science Thesis adviser: Dr. Joseph M. Prahl Department of Mechanical and Aerospace Engineering CASE WESTERN RESERVE UNIVERSITY January, 2012

2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis/dissertation of Matej Znidar i Master of Science candidate for the degree *. Dr. Joseph Prahl (signed) (chair of the committee) Dr. Jaikrishnan Kadambi Dr. Paul Barnhart 12/6/2011 (date) *We also certify that written approval has been obtained for any proprietary material contained therein.

3 To my father, Dr. Dobroslav Znidarčić, who inspired me to become an engineer and encouraged me to never stop learning.

4 Table of Contents List of Tables... ii List of Figures... iii List of symbols... v Acknowledgements... vii Abstract... viii 1 Introduction Axial flow compressors Multiple Circular Arc blades Compressor Design Program and the layout cone Description of the Compressor Design Program CDP Coordinate System Blade elements in the conic coordinate system Conformal mapping of a circular arc to the conic coordinate system Conformal mapping of a Smith and Yeh blade to the Cartesian coordinate system Computer code development Blade generation Panel code Calculations and results Blade element parameters and simulation setup Results Discussion Conclusions Appendix A: Description of a circular arc in the plane Appendix B: Description of the bending moment modification in CDP References i

5 List of Tables Table 1: Criteria to fully define a blade segment Table 2: CDP output used to generate a blade Table 3: Defining properties of the DCA sample blade from CDP Table 4: Blade properties used to generate the DCA sample blade on the plane Table 5: DCA cascade simulated conditions Table 6: Lift coefficient and error for the three layout methods Table 7: Input parameters required for defining an MCA blade in the plane ii

6 List of Figures Figure 1: Blade schematic with blade and flow angles labeled Figure 2: Schematic shock structure around a cascade of airfoils... 4 Figure 3: Schematic blade element centerline of a NASA MCA blade Figure 4: Comparison of total pressure loss coefficient between a DCA and MCA stator 6 Figure 5: Unwrapping the cone Figure 6: Blade element definitions on the unwrapped cone Figure 7: Blade angle definition for a circular arc in Cartesian coordinates Figure 8: Projection of a blade shape to the layout plane in the Smith and Yeh layout method Figure 9: Detail of the leading edge of a blade element Figure 10: Maximum thickness point of a blade element Figure 11: Blade section coordinate differences for the 76A first stage stator Figure 12: Schematic representation of the distribution of γ on the surface on an airfoil 29 Figure 13: Local panel coordinate system and dimensions Figure 14: Trailing edge panel showing the constant strength source and vortex Figure 15: Pressure coefficient over a Joukowski airfoil Figure 16: Pressure coefficient over the surface of a NACA 4412 airfoil in cascade Figure 17: Blade element profile in the transformed plane for all three layout methods. 38 Figure 18: Differences in blade shapes between different layout methods Figure 19: Difference in blade angle between a plane circular arc blade and blades laid out using the Crouse and Smith and Yeh layout methods in the transformed plane Figure 20: Dimensionless velocity on the surface of a blade in cascade iii

7 Figure 21: Detail of the dimensionless velocity near the leading edge Figure 22: Grid dependency check for a plane DCA blade Figure 23: Differences in coordinates for the suction and pressure surfaces for DCA blades on several cone half angles Figure 24: Dimensionless velocity over DCA airfoils on different layout cone angles Figure 25: Comparison between inviscid, theoretical velocity coefficient on a cascade of NACA 65-series airfoils to experimental results Figure 26: Rotor blade with labeled points iv

8 List of symbols A B C M R V b c f i l m n n p r s t u v v z α γ δ ε ζ η θ κ λ μ ξ ψ Influence coefficient due to vorticity Influence coefficient due to trailing edge source, number of blades Turning rate Number of nodes, moment Conic radius Velocity Boundary condition on the blade Chord Fraction of chord, function Incidence angle Moment arm Mass flow Normal coordinate Unit normal vector Pressure Cylindrical coordinate in the compressor, radial panel component Coordinate along the blade surface or centerline, tangential coordinate Blade thickness x-velocity component y-velocity component, velocity y-velocity component Axial coordinate in the compressor Conic half angle, flow angle of attack Vorticity density Deviation angle Conic angle Blade segment constant Circumferential lean of the blade stacking line, dummy Cartesian coordinate Circumferential coordinate in the compressor, tangential panel component Blade angle Axial lean of the blade stacking line Circle centers for Joukowski transform Dummy Cartesian coordinate, cascade stagger angle Dummy Cartesian coordinate Subscripts 0 Reference point 1 Inlet segment 2 Outlet segment A From node j B From node j+1 LE Leading edge PS Pressure surface v

9 SP SS TE c h j max n t trans Stacking point Suction surface Trailing edge Centerline Hub Index Maximum thickness point Normal direction Tangential direction Transition point vi

10 Acknowledgements The author would like to thank the Turbomachinery and Heat Transfer branch at NASA Glenn Research Center for their support. Without their financial and technical support, this research would not have been possible. The author would especially like to thank his mentor, Mr. John C. Fabian, whose assistance and encouragement proved invaluable throughout the project. Finally, the author would like to thank his thesis adviser, Dr. Joseph M. Prahl, for his support through the last several years. vii

11 Computational Validation of the Compressor Design Program Blade Layout Method Abstract by MATEJ ZNIDARČIĆ The blade layout method for circular arc blades used in the Compressor Design Program is compared to the layout method proposed by Smith and Yeh and to a circular arc in the plane. The blade is conformally transformed from the conic coordinate system used to lay out the blade to a Cartesian plane. The coordinates of blade elements in the transformed plane matched within 0.12% of the chord. The flow around the blades in cascade in the transformed plane is found using a vortex panel code. The lift coefficient on the blades for the three layout methods match within 2%. viii

12 1 Introduction 1.1 Axial flow compressors Modern gas turbine design for propulsion applications requires high efficiency operation in a compact and low weight engine. The need for high propulsive efficiency motivated the development of high pressure ratio core compressors. To achieve the high pressure ratios needed for efficient engine operation while also maintaining a low compressor mass and high compressor efficiency, high pressure ratio stages with high blade loading are required. Compressor stages achieve a total pressure rise by flow diffusion. Energy is added in the rotor, increasing the total pressure of the flow and the absolute flow velocity. Stator rows diffuse the flow, decreasing the absolute velocity and increasing the static pressure. In order to achieve design pressure ratios and design efficiencies, blade rows must be carefully designed to achieve the necessary flow turning while minimizing losses. In traditional design techniques, airfoil shapes are chosen from families with extensive experimental or theoretical performance estimates for two dimensional cascades. Empirical correlations are then used to relate cascade data to performance of the blade in a compressor. The open literature suggests that modern commercial compressor designs use blades that are modifications of proprietary blade families. However, research compressors still make use of traditional airfoil families that have been developed and described in the open literature. 1

13 Figure 1: Blade schematic with blade and flow angles labeled. For a compressor to achieve its design pressure ratio and efficiency, the flow turning and pressure losses must be adequately predicted. For an ideal blade row, pressure rise can be found by assuming constant enthalpy (for stators) or rothalpy (for rotors) and finding the flow turning through the blade row. Flow turning is found using the inlet and outlet blade angles and the incidence and deviation angles for a blade element (see Figure 1). With all four of those parameters specified, the velocity triangles at the inlet and outlet are known and the ideal pressure rise and flow turning achieved in a blade row is known. With known velocity triangles at the leading and trailing edges of the blade, the pressure rise in the blade row is known, even if the exact blade element geometry is not specified. However, an accurate estimate of the incidence and deviation angle is predicated on knowing the blade geometry. Blades with identical inlet and outlet angles that have different geometries will not, in general, have equivalent losses. Primary losses in high speed compressor blade rows can be divided into two categories: profile loss and shock loss. Profile losses are losses due to viscous effects, 2

14 including pressure losses due to skin friction, flow separation, and the deviation angle at the trailing edge of the blade. Shock losses are the losses in total pressure due to shocks at the leading edge of the blade row and in the blade-to-blade passage. The primary losses in a compressor blade row are determined from the blade profiles. 1.2 Multiple Circular Arc blades Blade elements are designed to achieve the necessary flow turning while minimizing losses. Early compressor designs for low solidity, high aspect ratio, low pressure ratio blade rows used conventional airfoil family blades, such as NACA four digit airfoils. Eventually, the need for higher pressure ratios motivated the use of double circular arc (DCA) or NACA-65 series airfoils sections for low speed applications. As compressor speeds increased to the transonic range, blade rows utilizing DCA or NACA- 65 series airfoils demonstrated relatively high losses due to the development of shocks at the blade leading edges and in the blade-to-blade passage. Multiple circular arc (MCA) blade shapes are a family of airfoils developed by NASA in the 1960s to address the need for efficient, high-speed compressor designs [1]. 3

15 Figure 2: Schematic shock structure around a cascade of airfoils (from Schwenk, Lewis and Hartmann). Blades in cascade at transonic speeds exhibit a shock pattern similar to Figure 2. A bow shock forms off the blunt leading edge. The fluid behind the shock expands over the suction surface to supersonic speed before terminating at a near-normal shock at some point on the approach to the blade-to-blade covered passage. Total pressure losses due to the shock in the blade passage are related to the strength of the shock, and therefore the Mach number immediately upstream of the shock. To limit these pressure losses, the strength of the normal shock should be reduced. In MCA blade shapes, this is accomplished by limiting the turning of the inlet section of the blade. The Mach number of a supersonic flow being turned through an angle can be found by utilizing the Prandtl-Meyer function. A flow expanded over the suction surface of a blade will accelerate, increasing the Mach number downstream of the leading edge. To limit the downstream Mach number, MCA blades limit the flow turning before the normal shock, effectively reducing the strength of the passage shock. After the shock, the blade turning is increased to achieve the total turning necessary for the compressor design. 4

16 Figure 3: Schematic blade element centerline of a NASA MCA blade. The transition point is marked in red. MCA shapes are defined by a centerline and a thickness distribution. In NASA s MCA blade family, the centerline consists of two circular arcs of different curvature, meeting at a transition point. The blade angle is continuous at the transition point. The thickness distribution can be described by the location of the maximum thickness point, the blade thickness at that point, and the leading and trailing edge thickness. The pressure and suction surfaces are each defined by two circular, matching the specified thickness at the leading and trailing edge and the maximum thickness point, and matching blade angle at the transition point. MCA shapes are a necessary compromise between shock and profile losses in transonic compressors. At design speeds, DCA or NACA 65-series airfoils achieve a given turning with minimal profile losses for subsonic compressor operation. In transonic or supersonic compressors designed with such blades, shock losses are large. Testing by Monsarrat and Keenan demonstrated that MCA shapes are effective at reducing the total pressure losses over DCA shapes in a transonic stator. Calculated profile losses are on the 5

17 same order for DCA and MCA shapes, indicating a substantial reduction in shock loss for the MCA shape with minimal, if any, increase in profile loss. The total pressure loss coefficient from Monsarrat and Keenan as a function of span is reproduced in Figure 4. Figure 4: Comparison of total pressure loss coefficient between a DCA and MCA stator (from Monsarrat and Keenan). MCA blade shapes continue to be of interest to researchers. Recent experimental studies about the shock structure around MCA blades in cascade have been presented by Šimurda, Luxa and Šafařík [2]. Rotor 37 is a rotor designed in the 1970s by NASA Lewis Research Center (now NASA Glenn Research Center) as part of a program to examine the flow in a transonic compressor inlet stage for various design parameters. Rotor 37 was designed with MCA blade shapes, and, in the early 1990s, it was used by the International Gas Turbine Institute as a test case to examine the use of CFD to predict turbomachinery performance [3]. Boretti, in comparing recent CFD results to experimental data, notes that despite Rotor 37 s older design, its performance is comparable to that of modern compressor rotor [4]. Several research front block core compressors developed by NASA, including the 74 and 76 compressor families and research inlet stages 35 through 38 were designed using MCA blade elements calculated by the Compressor Design Program (CDP) 6

18 developed primarily by Crouse [5]. The layout method used by Crouse to simulate the MCA blades is unique. Crouse, in following the method proposed by Crouse, Janetzke and Schwirian [6], lays out the MCA blades on axisymmetric streamlines approximated as cones using the fundamental characteristic of an arc, namely, that the curvature is constant. A review of the open literature did not reveal a rationale behind the choice of layout methods. Crouse notes in a personal correspondence that the choice of constant curvature on the cone was chosen to match the flow turning on the streamline to a circular arc in a plane [7]. However, no rigorous theoretical or experimental verification of this layout method was found. The objective of the present research is to determine the validity of the blade layout method used in CDP. The method is evaluated theoretically and compared to another layout method found in the open literature. 7

19 2 Compressor Design Program and the layout cone 2.1 Description of the Compressor Design Program CDP is a quasi-two dimensional throughflow compressor analysis code that determines the performance and blade design of multistage axial flow compressors. The code was written in several iterations starting in the mid-1970s with the last documented version by Crouse and Gorrell dating from 1981 [8]. CDP calculates velocity triangles at approximately radial stations located at the leading and trailing edges of each blade row and at annular stations within the flow passage. Flow properties at each station are calculated on discrete streamlines along the span of the blade. Streamlines are approximated by cones within the blade rows, and the coordinates for multiple circular arc blades that satisfy the flow turning requirements are calculated on the unwrapped cones. The blade elements are laid out on the cone using the method of Crouse, Janetzke and Schwirian. As input, CDP takes overall quantities for compressor performance, including design pressure ratio, rotational speed, and mass flow; the flowpath coordinates; and approximate locations of blade rows. Each blade row requires additional input to define the amount of flow turning and, therefore, the blade design. Additional input for blade rows includes the solidity; the number of blades; the ratio of turning rates of the blade inlet and outlet segments; incidence and deviation angles; the location of the transition point; the location and thickness of the maximum thickness points; and coefficients for the leading and trailing edge ellipses. Rotors have one required additional input to define the energy added by each rotor and an optional input to define the density of the blade material for simple mechanical design purposes. 8

20 Many of the compressor inputs need to be known in order to achieve the desired pressure ratio and efficiency. Because of this, considerable design effort is needed before using blade design features of CDP. For instance, the flow path, design speed and design mass flow rate must be accurately specified through one dimensional sizing analysis. CDP is, however, capable of calculating certain values important to the blade design. Incidence and deviation angles can be found from empirical correlations. The transition point location on a blade element can be either explicitly input or placed at the calculated location of the shock on the suction surface. The inlet-to-outlet segment turning rate ratio can be either input or determined by an empirical function of inlet Mach number. Many of the empirical models used within CDP come from NASA- SP-36, the standard reference for compressor design from that era. The flow in the blade-to-blade passage is not calculated. Because of this, appropriate loss models must be applied at the trailing edge calculation station. Profile losses are calculated based on a correlation to diffusion factor that is input by the user. Shock losses are handled by a modification of the loss model used by Schwenk, Lewis and Hartmann [9]. Schwenk et al. note that a good estimate of shock loss is necessary to predict the performance of multi-stage compressors since the existence of shocks in a blade row strongly affects the performance of downstream blade rows. The shock loss model presented by Schwenk et al. estimates the loss as the total pressure drop due to a normal shock at the entrance to the blade-to-blade channel. Crouse and Gorrell note that a normal shock cannot generally be sustained in the channel either because of shockboundary layer interaction or because an oblique shock would develop from the leading edge of an adjacent blade; experimental data (such as that presented by Šimurda et al.) 9

21 confirms that the shock loss in a blade passage is somewhat reduced from the loss due to a normal shock. To account for this reduction of shock losses, CDP reduces the total pressure loss of the passage shock by a factor of 1/M sh, where M sh is the Mach number of the flow just upstream of the shock. Crouse notes that no effort was made to validate this choice for the reduction of shock loss either theoretically or experimentally. Both profile and shock loss models are located in one subroutine and can be modified with minimal effort. To account for boundary layer growth at the hub and casing of the compressor, a blockage factor for each calculation station is input by the user. Other end wall effects are not explicitly accounted for in the code. After the aerodynamic design of each blade element, the elements are stacked such that a stacking point, defined as the individual element s center of area, is coincident with a stacking line that emanates from a stacking point on the hub. The stacking line can be tilted in both the axial and circumferential directions. For rotors, if the material density of the blades is input, CDP will calculate a circumferential and axial tilt that minimizes the bending moment at the blade root by counterbalancing the aerodynamic forces on the blade with the centrifugal force due to rotor rotation. CDP can output manufacturing coordinates of the compressor blades. These coordinates are defined at horizontal cuts through the blade and are found by interpolating between blade elements on the cone using a cubic spline. While CDP calculates some mechanical properties for the blade sections (such as the center of gravity, moments of inertia and twist stiffness), the mechanical design of the blades cannot be confirmed in CDP. Furthermore, the blade coordinates output by CDP are for the compressor running at the design point; any untwist of the blade necessary for manufacturing needs to be separately calculated. 10

22 2.2 CDP Coordinate System CDP lays out blade elements by unwrapping the cone it uses to represent a streamline in a blade row (Figure 5). The conic coordinate system is defined on this unwrapped cone. For a cone of half-angle α, the R coordinate of a given point on the unwrapped cone is given by R = r sin α (2.1) where r is the cylindrical coordinate of the point. The angular coordinate ε is obtained by observing that the perimeter of a circle swept out by a radius R on the cone must be equal to the perimeter swept out by the radius r at the same point. Thus, Rε = rθ ε = θ sin α (2.2) Since the cylindrical coordinates must fall on the cone for these relationships to be valid, z is constrained to z = r tan α. Figure 5: Unwrapping the cone 11

23 The (R, ε) coordinates can be visualized on a sector of a circle of angle 2π sin α. Within the context of CDP, the reference angle is defined as ε = 0 at the centerline transition point of the defined blade. The coordinate system for a sample blade element is shown in Figure Blade elements in the conic coordinate system The layout method used in CDP is unusual, and a review of the literature did not reveal a rigorous derivation or rationale behind it. The fundamental characteristic of a circular arc is that the change in blade angle along the length of the arc, dκ, is constant. CDP extends this characteristic to a cone by keeping the change in blade angle with respect to the radial line from the vertex of the cone constant (Figure 6). Crouse, Janetzke and Schwirian compared dκ on a cone for various layout methods and noted that ds projecting a circular arc to a cone using a variety of methods will not result in a constant dκ ds in the conic system. Despite this, Seyler and Smith [10], in an MCA blade design that predates CDP, used a layout method suggested by Smith and Yeh [11]; namely, the blade is laid out using a projection of the MCA shape from a plane to the streamline. Experimental data for a single stage from Gostelow et al. indicates that the method used by Seyler and Smith underpredicts mass flow, pressure ratio and efficiency at the design speed [12]. Data from Steinke on the multistage 74A designed using CDP compressor indicate that CDP underpredicts mass flow and overpredicts pressure ratio for the compressor at design speed [13]. Neither CDP nor the Smith and Yeh method provide for adequate surge margin at the design point. Because of the differences in the compressor designs, the layout methods cannot be directly compared using this experimental data. ds 12

24 Figure 6: Blade element definitions on the unwrapped cone (from Crouse, Janetzke and Schwirian) In order to understand the differences in the two layout methods, the methods were investigated. In Cartesian coordinates, the blade element centerline for a multiple circular arc blade is defined by two arcs of constant turning rate joined at a common transition point. Each segment is defined by dκ ds = C (2.3) where κ is the angle shown in Figure 7 and s is the distance along the arc. The surfaces are defined in the same way with different turning rates. Figure 7: Blade angle definition for a circular arc in Cartesian coordinates 13

25 Crouse et al. extend equation (2.3) to the cone by assuming the turning rate is also constant in conic coordinates. Using the geometry of the blade on the unwrapped cone (Figure 6), equation (2.3) can be expressed in (R, ε). ds = dκ C dr = cos κ ds Rdε = sin κ ds dr = cos κ dκ (2.4) C sin κ dε = RC dκ (2.5) The R coordinate can be found by direct integration of equation (2.4), giving R 1 R 2 = 1 C (sin κ 2 sin κ 1 ) (2.6) where the subscripts 1 and 2 refer to any two points on a given segment. Rearranging, CR 1 + sin κ 1 = CR 2 + sin κ 2 This indicates that the value ζ = RC + sin κ (2.7) is constant for a given segment. Thus, the R coordinate for any point on a given circular arc segment can be expressed in terms of the constants ζ and C, as well as the local blade angle κ: R = ζ sin κ C (2.8) This expression for R can be used in equation (2.5) to give dε = sin κ dκ (2.9) sin κ ζ 14

26 The integration of equation (2.9) depends on the value of the parameter ζ and can be expressed in general as ε = ε 1 + f(ζ, κ) 2.4 Conformal mapping of a circular arc to the conic coordinate system Since the unique layout method used in CDP was not rigorously derived or supported in the literature, the theoretical validity of the assumed blade shapes in CDP is investigated. If a circular arc airfoil is laid out on the conic system, and if it is correlated to a circular arc in a two-dimensional cascade (with blades laid out in a Cartesian system), the arc in conic coordinates must be conformally related to the arc in Cartesian coordinates. Since there is no closed form expression in (r, ε) coordinates for a circular arc blade generated by CDP, the inverse question is asked; namely, would a circular arc in Cartesian coordinates result in the relationship dκ = C when conformally mapped to conic coordinates? In Cartesian coordinates, a circle, centered around (x 0, y 0 ) is defined by the equation (x x 0 ) 2 + (y y 0 ) 2 = C 2 (2.10) Where C is the radius of the circle. For the similar transformation from Cartesian to Polar coordinates, Wislicenus [14] suggests a conformal transformation using the rules x = ln R (2.11) y = ε (2.12) ds 15

27 Thus, a circular arc mapped to conic coordinates can be expressed as (ln R ln R 0 ) 2 + (ε ε 0 ) 2 = C 2 R = R 0 exp C 2 (ε ε 0 ) 2 (2.13) To determine the change in blade angle along this curve, it is first necessary to define an appropriate slope. The conic coordinates are temporarily expressed in Cartesian coordinates. η = R cos ε (2.14) ξ = R sin ε (2.15) The tangent of an arbitrary curve is given by dξ. By inspection of Figure 6, this is equivalent, in conic coordinates, to dη dξ = tan (κ + ε) (2.16) dη To find dξ, equations (2.14) and (2.15) are differentiated with respect to ε. Noting that dη R = f(ε), Combining equations (2.17) and (2.18), dη dε = f (ε) cos ε f(ε) sin ε (2.17) dξ dε = f (ε) sin ε + f(ε) cos ε (2.18) dξ dε dε dη = dξ dη = f (ε) sin ε + f(ε) cos ε f (ε) cos ε f(ε) sin ε (2.19) From equation (2.16) and (2.19), the tangent of the local blade angle can be expressed as 16

28 dr sin ε + R cos ε tan(κ + ε) = dε (2.20) dr dε cos ε R sin ε dr dε = R 0(ε ε 0 ) C 2 (ε ε 0 ) exp 2 C2 (ε ε 0 ) 2 (2.21) Combining equations (2.20) and (2.21) and simplifying, ε ε 0 sin ε cos ε C tan(κ + ε) 2 (ε ε 0 ) 2 = ε ε 0 cos ε + sin ε C 2 (ε ε 0 ) 2 The blade angle, as a function of the conic angle ε, is given by ε ε 0 sin ε cos ε C 2 (ε ε 0 ) 2 κ = atan ε ε ε (2.22) 0 cos ε + sin ε C 2 (ε ε 0 ) 2 To find the variation of κ along the blade, a surface element must be defined. From Figure 6, it can be seen that Rearranging, ds 2 = dr 2 + R 2 dε 2 ds dε 2 = dr dε 2 + R 2 (2.23) With R and dr known from equations (2.13) and (2.21), respectively, an dε expression can be found for dκ ds using From the derivative of equation (2.22), dκ ds = dκ dε dε ds dκ dε = 1 C 2 (ε 0 ε) 2 17

29 and the inverse of ds that is implied by equation (2.23), the change in blade angle along dε the surface of a conformally mapped circle is given by dκ ds = 1 RC (2.24) The inverse relationship between the change in blade angle along the arc and the local conic radius indicates that the Crouse layout method is not a conformally mapped circular arc on the cone. By inspection of equation (2.24), it can be seen that for a given arc length, s, the change in dκ between the inlet and outlet of the arc will be small for ds large R. This condition corresponds to a small layout cone half angle. However, for typical layout cone angles near the hub and tip of an axial compressor, the variation in blade curvature on the cone will be on the order of 10% across the blade. 2.5 Conformal mapping of a Smith and Yeh blade to the Cartesian coordinate system The Smith and Yeh method is also compared to a circular arc conformally mapped to the conic coordinate system. Smith and Yeh suggest laying out a blade by projecting lines from the blade element in a plane to the streamline. The projecting lines are perpendicular to the plane and parallel to the axis of the blade. While Smith and Yeh suggested this method for laying out blades in a two dimensional cascade, they note that it is also valid for a blade with variations in blade element shape along its span. In investigating layout methods on the cone, Crouse, Janetzke and Schwirian extended this method to conic coordinates. On the cone, the blade is laid out by extending lines perpendicular to the layout plane to the intersection of the wrapped cone. This layout 18

30 method is shown schematically in Figure 8. Crouse et al. explicitly reject this method since it yields value of dκ that is not constant on the unwrapped cone. ds Figure 8: Projection of a blade shape to the layout plane in the Smith and Yeh layout method. The path of the intersection on the wrapped cone has a corresponding shape on the unwrapped cone, which in turn can be conformally mapped to a plane. The conic coordinates of a blade element can be expressed in three dimensional Euclidean space by the relations η = R sin α cos ε (2.25) sin α ξ = R sin α sin ε (2.26) sin α ψ = R cos α (2.27) To check the validity of the Smith and Yeh layout method, the conformally mapped circle in conic coordinates is expressed in (η, ξ, ψ) coordinates and projected to the ξ ψ plane. Substituting equation (2.13) into equations (2.26) and (2.27), ξ = R 0 exp C 2 (ε ε 0 ) 2 sin α sin ε (2.28) sin α ψ = R 0 exp C 2 (ε ε 0 ) 2 cos α (2.29) The tangent of blade angle with respect to the ψ axis is 19

31 tan κ = dψ dξ tan κ = dψ dε dε dξ (2.30) Taking the derivatives and simplifying, tan κ = C2 (ε ε 0 ) 2 cos( ε sin α ) (ε ε 0) sin( ε sin α ) sin α cos α (ε ε 0 ) (2.31) Instead of determining dκ for the projected circle, the expression for the local ds tangent line can be compared to that of a circle drawn in the ξ ψ plane. The tangent of the blade angle of a circle in the ξ ψ plane is ξ 0 ξ tan κ = C 2 (ξ ξ) 2 Expressing this in terms of the conic coordinates, tan κ R 0 sin α exp C 2 (ε ε 0 ) 2 sin ε = sin α exp(c) sin ε 0 sin α C 2 R 2 0 sin 2 α exp C 2 (ε ε 0 ) 2 sin ε sin α exp(c) sin ε 0 sin α 2 (2.32) Comparing equations (2.31) and (2.32), it is apparent that the two expressions for tan κ are not equivalent and, therefore, that the Smith and Yeh layout method also does not yield a conformally mapped circle on the unwrapped cone. For two dimensional cascades, Smith and Yeh support their layout method using physical reasoning; however, it is not readily apparent if the same reasoning applies to an annular cascade with axisymmetric streamlines. Likewise, Crouse et al. did not rigorously support the contention that dκ = C is the preferred layout method for an MCA or DCA ds 20

32 blade in a compressor. Since neither the Smith and Yeh method nor the Crouse method yield a conformally mapped circular arc on the cone, it is not clear which method is preferable for laying out an MCA or DCA blade in CDP. The uncertainty motivated the development of a method to compare the flowfield around blade elements laid out using the two methods. 21

33 3 Computer code development 3.1 Blade generation CDP outputs blade coordinates for manufacturing on horizontal sections through the blade; it does not output the blade coordinates on the cone. Since the comparison of the present study is for blade elements defined on the cone, a code was written to calculate the blade coordinates on the unwrapped cone using output available from CDP. An MCA blade laid out using the Crouse method consists of six individual segments. The centerline, suction and pressure surfaces each consist of an inlet and an outlet segment and each segment satisfies equation (3.1). dκ ds = C i (3.1) i A segment turning rate can be defined by any of the criteria listed in Table 1. The blade generator code uses output from CDP listed in Table 2 to calculate the turning rates and coordinates of each segment. CDP does not output turning rates; Crouse, Janetzke and Schwirian note that the blade shape is highly sensitive to variations in C. Due to rounding errors, specifying C could result in errors in the blade coordinates, so C is calculated rather than specified. Table 1: Criteria to fully define a blade segment Criteria C, R 1, κ 1 C, s, R 1 R 1, R 2, κ 1, κ 2 Description Turning rate, radius and blade angle at a point Turning rate, arc length, radius at a point Radius and blade angles at two points 22

34 Table 2: CDP output used to generate a blade CDP Output c (r, z) LE (r, z) TE t LE t TE f max t max κ LE κ TE κ trans C 1 /C 2 r sp,h λ η Description Chord Leading edge coordinates in (r, θ, z) coordinates Trailing edge coordinates in (r, θ, z) coordinates Leading edge thickness as a fraction of the chord Trailing edge thickness as a fraction of the chord Maximum thickness point location as a fraction of the chord Maximum thickness point thickness as a fraction of the chord Centerline leading edge blade angle Centerline trailing edge blade angle Centerline transition point blade angle Inlet to outlet segment turning rate ratio Hub stacking point r coordinate Stacking line lean angle in the (r, z) plane Stacking line lean angle in the (r, θ) plane The output available from CDP requires the centerline of the blade to be defined. The centerline transition point is found by first rearranging (2.6). R 1 R 2 = 1 C (sin κ 2 sin κ 1 ) (2.6) C = sin κ 2 sin κ 1 R 1 R 2 (3.2) Equation (3.2) is valid for any segment on the MCA blade. The turning rate ratio can be expressed as the quotient of the turning rates on the inlet and outlet segment. On the centerline, this is C 1,c = (sin κ LE sin κ trans )(R TE R trans ) C 2,c (R trans R LE )(sin κ trans sin κ TE ) (3.3) Rearranging equation (3.3) yields R TE R trans R trans R LE = C 1,c C 2,c sin κ trans sin κ TE sin κ LE sin κ trans (3.4) 23

35 The values on the right hand side of equation (3.4) are all known from input. For notational simplicity, the right hand side is denoted as D and the equation is solved for the transition point radius. R trans = DR LE + R TE 1 + D (3.5) The ε coordinate is, by definition, 0 at the transition point. With the transition point known, the turning rates for both the inlet and outlet segment can be found using equation (3.2). The value of ζ, defined in equation (2.7), for both segments is now known, and thus the coordinates of any point on the centerline can be found. Figure 9: Detail of the leading edge of a blade element. Surface tangency points are denoted with red marks; the line connecting the points is normal to the centerline leading edge blade angle. To find the coordinates of the suction and pressure surfaces, blade angles and coordinates of points on the surfaces are needed. The suction and pressure surface tangency points on the leading and trailing edge circles are found using a line normal to the centerline at the leading and trailing edges, shown in Figure 9. To find the turning rate of a segment on the blade surface, the blade angle and coordinates of one additional point on the segment need to be found. At the maximum thickness point, the blade angles at the centerline and on the suction and pressure surfaces are equivalent. The coordinates of the 24

36 maximum thickness point on the centerline are found using the intersection of a line normal to the chord line at the inputted fraction of the chord and the centerline. Since the centerline is not defined by an analytical equation, the coordinates are found using an iterative procedure. When the maximum thickness point on the centerline is found, the equivalent points on the pressure and suction surfaces are found by extending a path normal to the centerline at the maximum thickness point over a distance equal to the inputted maximum thickness (see Figure 10). Figure 10: Maximum thickness point of a blade element. The red line connects the maximum thickness point on the pressure and suction surfaces, and is normal to the blade angle at the maximum thickness point on the centerline. Since the maximum thickness point is not, in general, at the transition point, only the segment of the surface containing the maximum thickness point is now fully defined. To fully define the other segment, the coordinates and blade angle of the transition point are needed. The transition point on the pressure and suction surfaces are located on a line that is normal to the transition point on the centerline. The coordinates and blade angle of this point on the surfaces can be found using an iterative procedure using the turning rate and coordinates of the known surface segment. The transition point coordinates and blade angle found from this procedure are used to fully define the other surface segment, thereby fully defining the surface of the blade element on the cone. After all the blade elements are defined for a blade, the elements are stacked. During the stacking procedure, the blade element centers of area are moved to be 25

37 coincident with a stacking line that is defined in input. The stacking line for the blade is defined in (r, θ, z) coordinates, and can lean in both the (r, θ) and (r, z) planes. In CDP, the lean is determined by balancing the bending stresses caused by aerodynamic and centrifugal forces at the blade root. Since the blade generator code does not perform aerodynamic calculations, the stacking line lean is input rather than calculated. The stacking point of each blade element is then placed on the correct radial location along the stacking line to define the blade. Accounting for the stacking line lean, the stacking point on a given blade element in cylindrical coordinates can be expressed as z sp = z sp,h + r sp r sp,h tan λ (3.6) θ sp θ sp,h = sin 1 r sp,h r sp tan η 1 + tan 2 η r 2 sp (1 + tan r 2 η) tan 2 η 1 sp,h (3.7) The stacking shift is first estimated by finding the center of area of each of the blade elements in (R, ε) coordinates. The blades are then each shifted in θ and z so that the element center of area is coincident to the stacking line. Since the centrifugal forces on rotor blades act on cylindrical sections of the blade, stacking is also performed on nominally cylindrical sections (CDP approximates the cylindrical sections as horizontal cuts through the blade). The stacking sections are found by interpolating across the blade elements and finding the blade shapes at the x values of the blade element stacking points. The centers of area of these blade sections are then determined and a new shift in θ and z needed for each section is calculated. The shift is applied to the blade elements and the process is repeated until the total shift falls below a predetermined tolerance. The manufacturing coordinates are then found using the same interpolation as is used in the 26

38 stacking procedure at inputted horizontal and axial cuts on the blade. A more detailed explanation of the stacking line in CDP is given in Appendix B. The blade generator was verified by comparing the manufacturing coordinates generated by CDP for the first stage stator of the 76A compressor with a blade defined using the blade generator. The differences on the blade are show in Figure 11. The manufacturing tolerances on the blade sections are ± inches. The blade sections calculated by the blade generator code match the CDP output to within these tolerances, indicating the blade elements defined by the blade generator are adequate representations of CDP geometry. The distribution of the error on the blade surfaces indicates that the primary cause of the error is likely a difference in the setting angle of the blade elements between the two codes. Coordinate difference (in.) 1 x Differences for blade sections, 74A first stage stator 7.1 in. PS 9.5 in. PS 9.0 in. PS in. PS in. PS in. PS in. PS 6.4 in. PS 9.5 in. SS 9.0 in. SS in. SS in. SS in. SS 7.1 in. SS in. SS 6.4 in. SS Z axis index Figure 11: Blade section coordinate differences for the 76A first stage stator at horizontal cuts CDP (and, by extension, the blade generator code) outputs the geometry of the blade at the design conditions. Compressor blades are manufactured with a certain amount of untwist to account for the centrifugal and aerodynamic stresses. The accuracy of the blade setting angles in the hardware during operation will be dependent on the 27

39 exact flow conditions and the accuracy of the mechanical calculations. Because of this physical reasoning and because the calculated coordinates from the blade generator were within manufacturing tolerances, no attempt was made to adjust the setting angles in the blade generator code. 3.2 Panel code To determine the flowfield around a cascade of airfoils, a vortex panel code was written. The code is a variation on the method suggested by Katz and Plotkin [15] using a linearly varying vortex distribution as the panel element. The value of the vorticity density, γ, at the common end points of adjacent panels are equal, and the boundary conditions for the flow are applied at the midpoint of the panel (shown schematically in Figure 12). The velocity induced by a panel j at any given point in the flow is expressed as u j = 1 2 π Δs n log r j+1 + s θ r j+1 θ j γ j+1 j (3.8) + (Δs s) θ j+1 θ j n log r j+1 r j γ j v j = 1 2 π Δs s log r j r j+1 + Δs n θ j+1 θ j γ j+1 (Δs s) log r j r j+1 + Δs (3.9) n θ j+1 θ j γ j with coordinates defined as in Figure

40 Figure 12: Schematic representation of the distribution of γ on the surface on an airfoil used in the panel code Figure 13: Local panel coordinate system and dimensions. For an airfoil with M nodes (M 1 panels) and a blunt trailing edge, the vorticity distribution over the airfoil is found by solving the matrix equation A γ + b = 0 for the vector γ, where A is an M + 2 M + 2 matrix of influence coefficients, specifying the influence of each value of γ at every panel; b is a vector of length M + 2 specifying the effect of the freestream velocity at each panel; and γ is a vector of length M + 2 with the distribution of vorticity on the airfoil. Since the vorticity distribution of each panel is defined by two values of γ and the common node of adjacent panels share 29

41 one value of γ, a system of M 1 panels will have M unknowns. The addition of a Kutta condition adds two unknowns and three equations, fixing the circulation around the airfoil. The boundary conditions for the flow are applied at a collocation point at the midpoint of each vortex panel. The boundary condition used is a Neumann condition requiring zero normal flow velocity at each collocation point. To satisfy this, the vector b is defined as the velocity normal to a panel due to the freestream flow. This boundary condition at panel j is given by b j = V n ȷ To determine the influence panel j has on panel i, the airfoil is rotated and translated such that panel j is parallel to the abscissa and the panel edge corresponding to γ j is coincident with the origin. The values for γ j and γ j+1 are set to unity and the induced velocity at the collocation point of panel i is calculated using equations (3.8) and (3.9). For every collocation point, the induced velocity due to γ j is given by the vector u A,j, and the induced velocity due to γ j+1 is given by the vector u B,j. The normal velocity induced at panel i by all the vortex panels is given by M 1 v n,i = u A,1 γ 1 + u B,M 1 γ M + u A,j + u B,j 1 γ j j=2 n ı (3.10) Thus, the influence coefficient due to the vortex at node j (2 < j < M) on panel i is given by A i,j = u A,j + u B,j 1 n ı (3.11) For the coefficients due to the first and last vortex element 30

42 A i,1 = u A,1 n ı (3.12) A i,m = u B,M 1 n ı (3.13) The Kutta condition utilized in the panel code depends on the type of trailing edge on the airfoil. For a sharp trailing edge, where one endpoint of panel M 1 and one endpoint of panel 1 are coincident, requiring γ M = γ 1 is a sufficient boundary condition. For a blunt trailing edge (the condition expected for turbomachinery blades), a modification of the condition suggested by Eppler and Somers is used [16]. An additional trailing edge panel is added between the end points of panels 1 and M. Constant strength vortex and source elements are placed on the trailing edge panel (see Figure 14). The boundary condition at the inside of the trailing edge panel specifies zero normal and tangential flow. Finally, the condition γ M = γ 1 is also specified. These three additional equations and two additional unknowns ensure that the flow off the trailing edge is smooth. The matrix equation solved by the code for an isolated airfoil can be expressed as A 1,1 A 1,2 A 1,M A 1,γTE A 1,σTE A 2,1 A 2,2 A 2,M A 2,γTE A 2,σTE A M 1,1 A M 1,2 A M 1,M A M 1,γTE A M 1,σTE A TE,1 A TE,2 B TE,1 B TE,2 1 0 A TE,M A TE,γTE A TE,σTE B TE,M B TE,γTE B TE,σTE γ 1 γ 2 γ M 1 γ M γ TE σ TE (3.14) = v n,1 v n,2 v n,m 1 v n,te v t,te 0 31

43 Figure 14: Trailing edge panel showing the constant strength source and vortex elements For a cascade of airfoils, the same procedure is used with small modifications. The coordinates and collocation points for every airfoil in the cascade are first found by shifting the coordinates of the central airfoil to the appropriate position based on the solidity and stagger angle of the cascade. The shift between an airfoil and the one above it is given by Δx = c cos ξ (3.15) σ Δy = c sin ξ (3.16) σ Each airfoil has a self-influence coefficient matrix, satisfies a Kutta condition at its own trailing edge, and influences every other airfoil in the cascade. For a total of B blades in cascade, the matrix equation for the cascade can be expressed in terms of submatrices as [A 1,1] [A 1,2 ] [A 1,B ] [γ 1 ] [b 1 ] [A 2,1 ] [A 2,2 ] [A 2,B ] [γ 2 ] = [b 2 ] (3.17) [A B,1 ] [A 1,1 ] A B,B [γ B ] [b B ] The diagonal terms are the self-induced coefficients for a given blade, and are expressed by equation (3.14). The remaining terms, expressed as A i,j, give the influence 32

44 coefficients of blade j on blade i. The expressions for the off-diagonal matrices are similar to equation (3.14), but do not include the condition γ M = γ 1. The influence coefficients are again found using equations (3.8), (3.9) and (3.11). For a cascade containing B airfoils each with M 1 panels, there are a total of B(M + 2) unknowns and B 2 (M + 2) 2 influence coefficients; however, many submatrices in the influence coefficient matrix of equation (3.17) are equal, allowing for somewhat fewer calculations. The panel code was validated for an isolated airfoil by comparing the analytical pressure distributions over a Joukowski airfoil with the pressure distribution calculated by the panel code. To verify the method of specifying the Kutta condition, the trailing edge of the airfoil in the panel code was truncated to produce a blunt trailing edge. A comparison of the analytical pressure distribution and the panel code pressure distribution is shown in Figure 15. The effect of increasing the number of panels is also shown; above a minimum of about 50 panels, the solution is relatively insensitive to the number of panels Pressure coefficient on a Joukowski airfoil 250 panels 100 panels 50 panels 20 panels Exact Cp x/c Figure 15: Pressure coefficient over a Joukowski airfoil; µ x =-.1, µ y =.1, α=5 degrees. 33

45 The flow in the cascade was verified by comparing to calculations performed by Katzoff, Finn and Laurence on a cascade of NACA 4412 airfoils [17]. A comparison of their results with the results of the present panel method is shown in Figure 16. It can be seen that the panel method converges rapidly and compares favorably for even a small number of blades in the simulated cascade Pressure coefficient over a cascade of NACA 4412 airfoils NACA RP-879 Isolated airfoil 3 airfoils 21 airfoils 41 airfoils -3 Cp x/c Figure 16: Pressure coefficient over the surface of a NACA 4412 airfoil in cascade; α=9.7 degrees, ξ=0 degrees, σ= All airfoils in the cascade used 160 panels; NACA RP-879 reported velocities at 16 points on the blade. Compressibility effects can be estimated using a Prandtl-Glauert compressibility correction, however the panel code does not implement this. Since the goal of the study of layout methods is to qualitative determine the validity of the simulated MCA blades used in CDP, the exact nature of the flow field around the cascade is unimportant; the important factor is the pressure coefficient on the blade relative to other layout methods, and this can be adequately compared for purely incompressible flow. The deviation angle off the blade is not explicitly accounted for in the panel code. The Kutta condition employed in the code produces smooth flow at the trailing edge in the direction normal to the trailing edge panel. A non-zero deviation angle can be 34