DEVELOPMENT OF A METHODOLOGY TO ESTIMATE AERO-PERFORMANCE AND AERO- OPERABILITY LIMITS OF A MULTISTAGE AXIAL FLOW COMPRESSOR FOR USE IN

Transcription

1 DEVELOPMENT OF A METHODOLOGY TO ESTIMATE AERO-PERFORMANCE AND AERO- OPERABILITY LIMITS OF A MULTISTAGE AXIAL FLOW COMPRESSOR FOR USE IN PRELIMINARY DESIGN by SAMEER KULKARNI Submitted in partial fulfillment of the requirements For the degree of Master of Science 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 _Sameer Kulkarni candidate for the _Aerospace Engineering (MS) degree *. (signed)_jaikrishnan R. Kadambi (chair of the committee) J. Iwan D. Alexander John J. Adamczyk Joseph M. Prahl Mark L. Celestina (date) _10/27/2011 *We also certify that written approval has been obtained for any proprietary material contained therein.

3 This thesis is dedicated to my mother, Sunanda.

4 Table of Contents Table of Contents... i List of Figures... iii Acknowledgements... vii List of Symbols... ix Abstract... xi 1 Introduction Overview Objective and Scope Details of the Compressor and CFD Model Major Compressor Features Computational Grid and CFD Code Features of the CFD Model Theory of Stage Stacking Definitions of Stage Performance Parameters Stage Stacking Procedure Constructing the Stage Performance Curves Blockage Applications of the Stage Stacking Procedure Performance Estimates at 97% Corrected Speed i

5 5.2 Estimation of the Minimum Corrected Speed Operability and Bleed Sensitivity Estimates Construction of Initial Guesses for Full Compressor Simulations Assessment of a Correlation for Stalling Stage Pressure Rise Background of the Correlation Results of the Correlation Connecting the Correlation Results to Flow Physics Captured by CFD Conclusions and Recommendations Conclusions Future Work A Blockage Derived From the Continuity Equation References ii

6 List of Figures Figure 2.1: Meridional view of the compressor Figure 2.2: Meridional view of the mesh Figure 2.3: Block diagram of the APNASA algorithm, adapted from Lopez et al. (2000)- 11 Figure 3.1: Identical inlet/exit stations away from a bleed Figure 3.2: Discrete inlet and exit stations flanking a bleed Figure 3.3: Stage 1 characteristic from simulation of stages 1-6 compared to that from full compressor Figure 3.4: Stage 1 performance curves at 6 different corrected speeds Figure 3.5: Stage 2 performance curves at 6 different corrected speeds Figure 3.6: Stage 3 performance curves at 6 different corrected speeds Figure 3.7: Stage 4 performance curves at 6 different corrected speeds Figure 3.8: Stage 5 performance curves at 4 different corrected speeds Figure 3.9: Stage 6 performance curves at 4 different corrected speeds Figure 3.10: Stage 10 inlet total pressure profile comparison Figure 3.11: Stage 10 performance curves choke-side defined Figure 3.12: Stage 10 performance curves choke-side further defined Figure 3.13: Stage 10 performance curves Figure 3.14: Stage 11 performance curves Figure 3.15: Stage 12 performance curves Figure 3.16: Stage 13 performance curves Figure 3.17: Stage 7 performance curves iii

7 Figure 3.18: Stage 8 performance curves Figure 3.19: Stage 9 performance curves Figure 4.1: Illustration of boundary layer displacement thickness within a compressor stage Figure 4.2: Stagewise blockage distributions at design point Figure 4.3: Stagewise blockage distributions at 100% speed near-stall point Figure 4.4: Stagewise blockage distributions at a part-speed near-stall point Figure 4.5: Stage 1 exit effective blockage curves Figure 4.6: Stage 5 exit effective blockage curve Figure 5.1: Stage stacking estimates along the stage 1 performance curve, 97% speed 57 Figure 5.2: Stage stacking performance estimate compared to CFD at 97% speed Figure 5.3: Stage 1 performance curves and stage stacking estimate at N2% speed Figure 5.4: Stage 9 performance curves and stage stacking estimate at N2% speed Figure 5.5: Stage 10 performance curves and stage stacking estimate at N2% speed Figure 5.6: Stage 1 performance curves and stage stacking estimate at N% speed Figure 5.7: Stage 13 performance curves and stage stacking estimate at N% speed Figure 5.8: Stage 1, N2% speed stage stacking estimate and CFD (increased bleed rate)66 Figure 5.9: Stage 9, N2% speed stage stacking estimate and CFD (increased bleed rate)67 Figure 5.10: Stage 13, N2% speed stage stacking estimate and CFD (increased bleed rate) Figure 6.1: Typical diffuser performance chart, adapted from Johnson (1998) Figure 6.2: Compressor blade geometry, adapted from Johnson (1998) iv

8 Figure 6.3: Correlation for stalling static pressure rise coefficient, adapted from Koch (1981) Figure 6.4: Effect of Reynolds number on stalling static pressure rise coefficient Figure 6.5: Effect of tip clearance on stalling pressure rise coefficient Figure 6.6: Effect of axial spacing on stalling pressure rise coefficient Figure 6.7: Correlation of static pressure rise coefficients for stages near design point 87 Figure 6.8: Correlation of static pressure rise coefficients for near-stall point at 100% speed Figure 6.9: Correlation of static pressure rise coefficients for near-stall point at N% speed Figure 6.10: Stalling pressure rise ratio - design point compared to 100% speed near-stall point Figure 6.11: Stalling pressure rise ratio - design point compared to N% speed near-stall point Figure 6.12: Rotor 5 tip leakage flow, near-stall points at 100% speed (left) and partspeed N% (right) Figure 6.13: Rotor 8 tip leakage flow, near-stall points at 100% speed (left) and partspeed N% (right) Figure 6.14: Rotor 12 tip leakage flow, near-stall points at 100% speed (left) and partspeed N% (right) Figure 7.1: Normalized total pressure profiles aft rotor 1, with and without fillet geometry v

9 Figure 7.2: N% speed, entropy contours at rotor 1 trailing edge, no fillets (left) vs. fillets (right) Figure 7.3: : N% speed, entropy contours at stator 1 trailing edge, no fillets (left) vs. fillets (right) Figure 7.4: : N% speed, entropy contours at rotor 2 trailing edge, no fillets (left) vs. fillets (right) Figure A.1: An element in the axisymmetric curvilinear coordinate system (ξ, η) Figure A.2: Streamlines defining the core flow region in a compressor stage vi

10 Acknowledgements I would first like to acknowledge that this research was funded by the NASA Graduate Student Researchers Program (GSRP) through a Reimbursable Space Act Agreement between NASA Glenn Research Center and Siemens Energy, Inc. I would like to thank Dr. Wai-Ming To and Mr. Richard Mulac of the University of Toledo for generously sharing their office space and computing resources at NASA Glenn Research Center, and especially for sharing their time with me in many technical discussions regarding turbomachinery and computational fluid dynamics (CFD). I would like to acknowledge several engineers of Siemens Energy, Inc. (SEI). I thank Dr. Matthew Montgomery, who offered this research opportunity and for his comments and suggestions during many technical discussions. I thank Mr. Elliot Griffin, who was my host during an internship with SEI, where I completed this research. I also thank Mr. David Wasdell, Dr. Eric Donahoo, and Dr. Christian Cornelius for many technical discussions and insights, and I further thank Dr. Donahoo for reviewing this thesis. I acknowledge Mr. Paul Spedaler for his assistance in providing the compressor geometries and in obtaining computing resources during my time at SEI. I would like to thank Dr. Choon Tan of the Massachusetts Institute of Technology (MIT) for his suggestions and remarks during many technical discussions, and for providing me with the opportunity to present a portion of this research at the MIT Gas Turbine Laboratory, which yielded helpful feedback from students and faculty. I acknowledge vii

11 the time and effort of my professors at Case Western Reserve University, Dr. Jaikrishan Kadambi, Dr. Iwan Alexander, and Dr. Joseph Prahl, for reviewing this thesis. I further thank Dr. Kadambi, my faculty advisor, for his support of this work. I give special thanks to my research advisors, Dr. Mark Celestina and Dr. John Adamczyk of NASA Glenn Research Center. I am grateful for the time and energy which they generously provided while guiding me through this research effort. Over the course of the work, their mentoring has allowed me to gain insight into CFD simulations and turbomachinery aerodynamics. This work would not have been possible without their support and advice. I further acknowledge them for providing access to and training on the CFD code used in this work, and for reviewing this thesis and helping to shaping it into its current form. Finally, I express my love and gratitude to my mother, Sunanda, and my sister, Mitali, for their everlasting support and encouragement. viii

12 List of Symbols ENGLISH SYMBOLS A an Geometrical annulus area, ft 2 A ef Effective flow area, ft 2 C Midspan airfoil chord length, ft C h C p c p F ef Stage static pressure rise coefficient Two-dimensional diffuser static pressure rise coefficient Specific heat at constant pressure, ft-lb f /(slug R) Effective dynamic pressure factor g c Gravitational constant = ft/sec 2 g Midspan staggered spacing between adjacent blades in a row, ft ΔH Change in total (stagnation) enthalpy, (ft/sec) 2 k b L M ṁ N N s N c Effective blockage factor Midspan arc length of circular-arc airfoil, ft Mach number Mass flow rate, lb m /sec Two-dimensional diffuser length, ft Shaft rotational speed, RPM Corrected rotational speed, RPM P Total (stagnation) pressure, lb f /ft 2 P sl NASA standard sea-level pressure = 2116 lb f /ft 2 p s Static pressure, lb f /ft 2 R Gas constant = ft-lb f /(slug R) r m S Midspan radius, ft Midspan pitch, ft T Total (stagnation) temperature, R T sl t Static temperature, R NASA standard sea-level temperature = R ix

13 U m V z V z Δz Midspan rotor speed, ft/sec Axial flow velocity in the absolute frame of reference, ft/sec Axial flow velocity in the relative frame of reference, ft/sec Midspan axial spacing between blade rows, ft GREEK SYMBOLS α Flow angle in the absolute frame of reference, deg β Flow angle in the relative frame of reference, deg γ Ratio of specific heats ε Rotor tip or stator hub clearance, ft η Adiabatic efficiency Λ Degree of reaction based on static pressure rise ρ Density, lb m /ft 3 Φ Midspan camber angle, deg φ Flow coefficient ψ Ideal work coefficient ω Shaft angular frequency, rad/sec SUBSCRIPTS 1 Inlet station of the stage, blade row, or diffuser 2 Exit station of the stage, blade row, or diffuser ref A reference value used for normalization SUPERSCRIPTS * Normalized value x

14 Development of a Methodology to Estimate Aero-Performance and Aero-Operability Limits of a Multistage Axial Flow Compressor for Use in Preliminary Design Abstract by SAMEER KULKARNI The preliminary design of multistage axial compressors in gas turbine engines is typically accomplished with mean-line methods. These methods, which rely on empirical correlations, estimate compressor performance well near the design point, but may become less reliable off-design. For land-based applications of gas turbine engines, offdesign performance estimates are becoming increasingly important, as turbine plant operators desire peaking or load-following capabilities and hot-day operability. The current work implements a one-dimensional stage stacking procedure, including a new blockage term, which is used to estimate off-design compressor performance and operability range of a 13-stage axial compressor used for power generation. The procedure utilizes stage characteristics which are constructed from computational fluid dynamics (CFD) simulations of groups of stages. The stage stacking estimates match well with CFD results. These CFD results are used to assess a metric which estimates the stall limiting stages. xi

15 1 Introduction 1.1 Overview Axial flow gas turbine engines have major roles in the aviation and power generation industries. In aviation applications, the most recognizable gas turbine engine is the jet engine, which provides thrust. In the power generation sector, industrial gas turbine engines are used to provide power to the electrical grid by burning fuels such as natural gas. The primary design goal of the jet engine is to maximize thrust while concurrently minimizing weight, whereas the primary design goal of the industrial gas turbine engine is to provide shaft horsepower at a high thermal efficiency. While the design goals for these engines differ, the flow physics involved in their operation is quite similar. Thus, insight into their flow physics is beneficial in understanding how to improve these engines, regardless of the end application. Gas turbine engines are described thermodynamically by the Brayton cycle. In this cycle, work is generated by compressing air isentropically, heating it in an isobaric process, and then expanding it isentropically back to its initial pressure. The cycle identifies the three major components of axial flow gas turbine engines: the compressor, the combustor, and the turbine. The challenges in designing each of these three components differ. For example, challenges in turbine design arise from the large range of temperatures and stresses experienced by the turbine blade components. These challenges are addressed via the use of secondary air flow to cool the blade surfaces, as well as the careful selection of materials used for the turbine blading. A 1

16 combustor design generally requires the capability to fully combust fuel while meeting pollutant emissions standards. In a compressor, which is the focus of the current work, the main challenge is to provide efficient designs which exhibit stable operation across a range of throttle conditions and rotating speeds, which demands the careful aerodynamic design of the compressor blading and flow path. Compressor performance is measured by the pressure and temperature rise across the compressor. For a given rotating shaft speed, the pressure rise will increase as mass flow through the compressor is decreased. However, there is a point, referred to as the stall point, at which the compressor reaches a maximum pressure rise. Any further reduction in mass flow results in one of two phenomena: rotating stall or surge (Cumpsty 1989). A compressor is said to be in rotating stall when regions of low axial velocity flow, called stall cells, form in the annulus and rotate in the direction of the rotor. If stall cells occupy the entire span of the annulus, the compressor is said to be in full-span stall. This leads to an abrupt stall of the compressor, characterized by a large drop in pressure rise and flow rate. If stall cells do not fill the entire span of the annulus, the compressor is in part-span stall, indicated by a gradual drop in overall performance due to a progressive stall of the compressor. Part-span stall is likely to develop into fullspan stall if the compressor is throttled further, which then causes an abrupt performance drop. The more severe phenomenon is the occurrence of surge, in which the mass flow through the entire engine reverses. This results in a total loss in pressure rise across the compressor and may additionally cause catastrophic damage to the 2

17 engine. Thus, identification of the stall point for a compressor is critical in determining the operability range of a gas turbine engine. In aviation applications, compressors in jet engines must have a wide operability range to account for diverse mission requirements and operating conditions. These compressors must be designed for stable operation at design speed, over-speed, and part-speed operation, in addition to having high efficiency during cruise. Jet engine compressors must operate at a range of altitudes, resulting in a variety of inlet air densities and temperatures which results in variation of Reynolds number, which in turn affects compressor performance. Compressors in industrial gas turbine engines used for power generation are designed to operate continuously at a physical shaft speed corresponding to the utility frequency of either 50 Hz or 60 Hz. Industrial engines generally operate continuously for months while generating a fixed power output, and only require inspection and maintenance every few years. Typical maintenance intervals for these baseload engines are on the order of 25,000 hours or 800 starts (Kiesow and McQuiggan 2007). Due to the operation of these engines at a fixed shaft speed, stability issues arise. The engine power output requirement is fixed, whereas the compressor s inlet temperature can vary according to the local climate, causing a change in the rotational Mach number. If the rotational Mach number is reduced from its design value, as on a hot day, the operating point of the compressor moves towards the stall boundary. Operation of an 3

18 engine in extreme climates, such as desert regions, results in appreciable changes in the corrected speed N c, which is defined in Eq N c = N s T T sl Eq. 1.1 Here N s is the physical shaft speed, T is the ambient, or compressor inlet total temperature, and T sl is the NASA standard sea-level temperature of R. Hot-day operation results in a reduction in the corrected speed, whereas cold-day operation increases the corrected speed. It has been reported that industrial gas turbine engine power output can drop by as much as 0.9% for every 1 C rise in ambient temperature (Chaker and Meher-Homji 2007). Care must be taken during the preliminary design of these compressors to ensure stable operation across the expected range of ambient temperatures. Besides the design requirements imposed by hot day operation, additional requirements are emerging due to the expected increase of renewable energy sources in the coming years (Kiesow and McQuiggan 2007). Wind turbines and solar cells will provide a varying amount of power dictated by local weather conditions. Considering this fact, it is becoming more economical for gas turbine plant operators to provide power during the hours when consumption demand, and thus the price of electricity, is highest. This large demand generally exceeds the capacity that can be met by baseload engines, wind turbines, and solar energy sources. The gas turbine engine requires a cyclic service duty to achieve this peaking capability. The future engine must be able to provide a range of power output at fixed wheel speed (as opposed to a fixed power 4

19 output at fixed wheel speed) so that the engine output can follow demand throughout the day. This effectively requires the engine to be designed for multiple design points and presents an engineering challenge in the aerodynamic design of the compressor. In a preliminary aerodynamic design system, empirically based methods, referred to as mean-line or through flow models, are used by engine manufacturers to establish an initial geometry for a new compressor (Glassman 1992, Hearsey 2007, Casey and Robinson 2010). These techniques make use of the breadth of experience and knowledge accrued from past compressor designs in the form of empirical or semiempirical correlations to estimate parameters of the flow, such as blockage, loss, and deviation, which have a large impact upon the performance and operability of the compressor. These mean-line or through flow models can estimate the performance of a new compressor design quite well near its design point. However, these methods can become less reliable off-design, or for new designs which differ substantially from previous ones. Off-design performance estimates are becoming more important, as hot-day operation and load-following or peaking capabilities become increasingly desirable for turbine plant operators. This poses a challenge during a preliminary design when critical performance estimates far from the design point are required. Thus, there exists a need for a methodology to generate credible estimates of off-design performance and operability of these compressors during preliminary design. 5

20 With the increasing power and availability of computer resources, engine manufacturers can now use three-dimensional computational fluid dynamics (CFD) solvers to examine off-design operation of a compressor. These solvers rely less on empirical correlations than most mean-line and throughflow methods. A properly validated and calibrated CFD solver can thus provide insight into off-design compressor performance. The flow information provided by the CFD simulations can be used to enhance the empirical correlations used by mean-line and throughflow codes. The CFD codes typically solve the Reynolds-averaged form of the Navier-Stokes equations numerically on a threedimensional computational grid which models the compressor geometry. Inlet and exit boundary conditions and shaft speed are specified to simulate conditions at various operating points. During the early stages of a compressor design, it is also of interest to compressor designers to understand how stall inception occurs at a given operating condition. When a compressor stage enters rotating stall, the stall is generally classified as either blade stall or wall stall. A characteristic of blade stall is a large region of low velocity fluid along a significant portion of the suction surface of the blade (Greitzer, et al. 1979). In wall stall, stall cells which are associated with the endwall boundary layer of either the hub or casing are formed. Depending on the type of stall, a designer may choose to redesign the compressor in order to extend the stall margin of the machine. For example, it was shown by Greitzer, et al. (1979) that casing treatments greatly improved the stall margin of a rotor exhibiting wall stall, but had little effect on the stall margin of 6

21 a rotor exhibiting blade stall. Therefore, there is a need to identify those flow features which lead to stall. A portion of the current work attempts to use CFD results to assess a metric which can identify stall associated with the endwall flow. In the current work, the compressor of an industrial gas turbine engine is considered. In general, middle and rear stages in these compressors have larger tip-clearance-to-span ratios as compared to aero engine compressors, due to the greater reduction in blade length through the compressor (Williams, et al. 2010). The relatively large tip clearances result in stronger tip leakage flows, which impact the flow in the endwall region. Thus, the focus of the current work was to assess a metric to identify wall stall, and metrics associated with blade stall were not assessed. Further information on typical metrics associated with blade stall can be found in Greitzer, et al. (1979) and Cumpsty (1989). 1.2 Objective and Scope The objective of this work is to develop a methodology which leads to credible estimates of aero-operability and aero-performance of multistage axial compressors at design and at off-design operating points for use in preliminary design. This objective is accomplished by developing a one-dimensional stage stacking procedure and using it to estimate design and off-design performance and operability limits of a 13-stage axial flow compressor. This compressor is part of an industrial gas turbine engine used for power generation. The stage stacking procedure utilizes stage performance curves, which are constructed using flow information extracted from a database of CFD simulations of the compressor. This database is generated by using the APNASA CFD 7

22 code for multistage turbomachinery. Finally, a metric for stage stalling pressure rise is applied and assessed using this CFD database. 8

23 2 Details of the Compressor and CFD Model 2.1 Major Compressor Features The focus of the current work is a 13-stage axial flow compressor which is part of an industrial gas turbine engine. The design shaft speed for the compressor is 3600 RPM, corresponding to a 60 Hz utility frequency. The compressor consists of an inlet guide vane (IGV), 13 rotor-stator stages, and an outlet guide vane (OGV). The IGV as well as stators 1, 2, and 3 (referred to as variable guide vanes, or VGVs) have variable stagger angles. The remaining stators and OGV are cantilevered. There are three casing bleeds in the compressor, located downstream of stators 5, 8, and 11. Compressor bleeds change the aerodynamic matching between blade rows. In industrial applications, bleeds are used to extract flow in order to unchoke the rear stages during engine startup, when the shaft speed is well below its design value. The bleed flow rates are reduced to nominal values as design shaft speed is approached. A portion of the flow extracted from the bleeds is used for cooling of the turbine section of the gas turbine engine. Excess flow extracted from the bleeds is expelled from the engine. An illustration of the meridional view of the compressor is shown in Figure 2.1. Figure 2.1: Meridional view of the compressor 9

24 2.2 Computational Grid and CFD Code A computational grid modeling the compressor was created using the mesh generator detailed by Mulac (1986). This generator produced an H-mesh for each blade row, with common axial and radial coordinates shared between the meshes. These blade row meshes contained regions overlapping the geometrical spaces of neighboring blade rows. This feature of the meshes was a requirement of the CFD code, discussed below. These meshes were generated for all 28 blade rows of the compressor, from IGV through OGV. The meshes contained 51 nodes in the tangential direction (from the suction side of one blade to the pressure side of the next blade) and 51 nodes in the radial direction (from hub to casing). Each blade contained 61 axial nodes along the chord, from leading edge to trailing edge. The number of axial nodes from the inlet plane to the exit plane of the grid totaled 3067 nodes. A meridional view of the numerical mesh is shown in Figure 2.2. Figure 2.2: Meridional view of the mesh Computational fluid dynamics simulations of the compressor were generated using the APNASA turbomachinery code. The APNASA code was developed by Adamczyk, et al. (1990) at NASA Glenn Research Center. The code solves a set of equations known as the 10

25 Average-Passage model, derived by Adamczyk (1985), which is based on a sequence of mathematical averaging of the Navier-Stokes equations, continuity equation, energy equation, and the equation of state. The Average-Passage model describes the timeaveraged flow field within a blade row embedded in a multistage environment, and attempts to account for the unsteady effects of neighboring blade rows by calculation of deterministic stresses carried through the momentum equation. In order to obtain these deterministic stresses, the code requires a computational domain for a particular blade row which overlaps the domains of neighboring blade rows. A block diagram of the APNASA algorithm, adapted from Lopez et al. (2000), is shown in Figure 2.3. Figure 2.3: Block diagram of the APNASA algorithm, adapted from Lopez et al. (2000) 11

26 The code employs a standard k-ε turbulence model to close on the Reynolds stress term. Details of the code algorithm and governing equations can be found in Adamczyk (1985), Adamczyk, et al. (1986), Adamczyk, et al. (1990), and Celestina (1999). 2.3 Features of the CFD Model The CFD model included all 28 blade rows of the compressor from IGV through OGV. The three VGVs were set to nominal settings, and the IGV was closed 2 deg from nominal to match the flow rate obtained in experimental tests. These setting angles were fixed irrespective of simulated operating conditions. Stator platform leakages associated with the IGV and VGVs were not modeled in this work. The three casing bleeds were modeled using an aspiration model which accounted for the mass flow rate removal along the casing at locations downstream of stators 5, 8, and 11. The bleed flow rates at design speed were set to their nominal values. At part-speed operation, bleed rates were scaled by percent speed. The physical geometry of the casing bleeds were not modeled in this work. An inlet boundary condition was prescribed at the first axial plane of the computational grid, at a location upstream of the IGV leading edge. This inlet boundary condition was specified as a radial profile of total pressure, total temperature, tangential flow angle, and radial flow angle. The inlet profile was obtained from an existing throughflow calculation for this machine, and was fixed irrespective of operating conditions. An exit boundary condition was prescribed at the last axial plane of the computational grid. This exit boundary condition was a static pressure specified at the hub, and simple radial 12

27 equilibrium was used to establish the spanwise pressure distribution (Adamczyk, et al. 1990). The simulation was throttled along a speed line by varying this exit hub static pressure. The corrected speed was varied by specifying the physical shaft speed, as opposed to changing the inlet temperature. This was done to emulate a series of surge tests that had previously been performed for this compressor. In these experiments, the IGV and VGV settings were fixed and mechanical shaft speed was incrementally reduced until the occurrence of a surge event. It should be noted that this operation differs from routine operation of the engine, in which corrected speed varies according to changes in ambient temperature conditions and the physical shaft speed of the compressor is fixed according to the utility frequency of 60 Hz. 13

28 3 Theory of Stage Stacking The first sections of this chapter detail a one-dimensional procedure for estimating overall compressor performance known as stage stacking, which is frequently used in the preliminary design of axial compressors. The procedure used in the current work was based on the method described by Robbins and Dugan (1965). The stage stacking procedure can be used to estimate design or off-design operating points if the individual stage performance curves, or characteristics, are known. These stage characteristics present performance of a stage as a function of its inlet equivalent mass flow and corrected speed. The last section of this chapter details a method to construct these stage performance curves using a multistage CFD solver. 3.1 Definitions of Stage Performance Parameters The stage stacking procedure of Robbins and Dugan (1965) is based upon definitions of non-dimensional stage performance parameters. These parameters consist of an ideal work coefficient and an adiabatic efficiency which are plotted as a function of stage inlet flow coefficient to form the stage performance curves. In the current work, these curves were generated using flow information extracted from CFD simulations, as will be detailed in Section 3.3. The stage ideal work coefficient ψ is defined as the ideal enthalpy rise across the stage, non-dimensionalized by midspan wheel speed squared, as shown in Eq

29 ψ = c p T sl P 2 P 1 T 1 γ avg 1 γ avg U m,1 T sl 2 1 Eq. 3.1 In the current work, total temperature T 1 and total pressures P 1 and P 2 are mass averaged values obtained from the CFD derived database detailed in Section 3.3. The subscripts 1 and 2 indicate the stage inlet and exit stations, respectively. The NASA standard sea-level total pressure and total temperature, P sl and T sl, are 2116 lb f /ft 2 and R, respectively. The ratio of specific heats used here, γ avg, is the arithmetic average of the ratios of specific heat at the stage inlet and stage exit stations, γ 1 and γ 2, respectively. The specific heat at constant pressure c p is given by Eq γ avg c p = R γ avg 1 Eq. 3.2 Here, R is the specific gas constant for air, ft-lb f /(slug R). The midspan rotor speed at the stage inlet station U m,1 is given by Eq U m,1 = ωr m,1 Eq. 3.3 Here, ω is the angular frequency of the rotating shaft with units of rad/sec and r m,1 is the midspan radius at the stage inlet station. Stage adiabatic efficiency η is defined as the ratio of the ideal total enthalpy rise to the actual total enthalpy rise, as shown in Eq

30 η = P 2 P 1 γ avg 1 γ avg 1 T 2 T 1 1 Eq. 3.4 Values of ψ and η are plotted as a function of stage inlet flow coefficient φ 1, which is defined as the ratio of corrected axial velocity to corrected midspan rotor speed, as in Eq φ 1 = V z,1 T 1 T sl U m,1 T 1 T sl Eq. 3.5 The corrected axial velocity used here is solved from the flow equation given in Eq m T 1 T sl A ef,1 P 1 P sl = V z,1 T 1 V z,1 1 T 1 T sl T sl 2 1 2c p T sl cos 2 α 1 1 γ 1 1 g c P sl RT sl Eq. 3.6 The design value of stage inlet midspan absolute flow angle α 1 is used here, and g c is the gravitational constant, ft/sec 2. Here ṁ is the mass flow rate at the stage inlet station. The value of stage inlet mass flow rate is equal to the compressor inlet mass flow rate, less any applicable flow removal due to bleeds. It should be noted that the current work expands upon the method described by Robbins and Dugan regarding the area term A ef,1 on the left-hand-side of Eq As noted by Robbins and Dugan, this term is generally taken to be the geometric annulus area. In the current work, the term A ef is taken as an effective flow area which attempts to include the effects of endwall flow blockage. A simple introductory definition for A ef is given by Eq A ef = A an [1 k b ] Eq

31 Here A an is the geometric annulus area and k b is a blockage factor. A method for calculating k b and hence A ef, as well as further discussion on blockage, is presented in Chapter 4. For the purposes of the stage stacking procedure, the stage 1 inlet station is defined as the first axial plane of the computational mesh, and the stage 1 exit station is at a location downstream of the trailing edge of stator 1. Thus, stage 1 includes the IGV, rotor 1, and stator 1. The stage 13 inlet station is at an axial location upstream of rotor 13, with its exit station coinciding with the last axial plane of the computational mesh, thereby including rotor 13, stator 13, and the OGV. Stages 2 through 12 are defined to include their respective rotor and stator pair. For stages away from the three casing bleed locations, the axial location defining the stage exit station is selected to be identical to the location of the inlet station of the downstream stage. This is illustrated in Figure 3.1, which depicts the single location of the stage 2 exit plane and stage 3 inlet plane. Figure 3.1: Identical inlet/exit stations away from a bleed 17

32 For stages 5, 8, and 11, which are directly upstream of a casing bleed, the stage exit stations are defined at locations downstream of the stator trailing edges, but upstream of the bleeds. Likewise, the inlet stations for stages 6, 9, and 12 are set to locations downstream of the bleeds, but upstream of the rotor leading edges. This is depicted in Figure 3.2, which shows the two distinct axial locations defining the stage 5 exit station and the stage 6 inlet station. Figure 3.2: Discrete inlet and exit stations flanking a bleed This was done so that flow information would not be extracted from a location within the bleed passage when calculating ψ, η, k b and φ 1. Curves of ψ, η, and k b as a function of φ 1 comprise the stage characteristics which are necessary in order to use the stage stacking procedure described in Section Stage Stacking Procedure The following must be known to estimate compressor performance using the current stage stacking procedure: 1. Fully populated performance curves for each stage at the corrected speed of interest: 18

33 o Stage ideal work coefficient vs. stage inlet flow coefficient (ψ vs. φ 1 ) o Stage adiabatic efficiency vs. stage inlet flow coefficient (η vs. φ 1 ) o Stage inlet effective blockage vs. stage inlet flow coefficient (k b,1 vs. φ 1 ) o Stage exit effective blockage vs. stage inlet flow coefficient (k b,2 vs. φ 1 ) 2. Geometrical annulus area at each stage inlet and exit station (A an,1 and A an,2 ) 3. Midspan radius at each stage inlet and exit station (r m,1 and r m,2 ) 4. Design value of midspan absolute flow angle at each stage inlet and exit station (α 1 and α 2 ) In the current work, the stage performance curves were generated using flow information extracted from a database of CFD simulations. The procedure for constructing these curves is detailed in Section 3.3. Performance plots for each stage must undergo a curve-fitting procedure to yield a line of best fit which excludes any obvious outliers. The equations defining these best-fit lines give ψ, η, k b,1, and k b,2 as a function of φ 1 for a given corrected speed. These equations are used in the current stage stacking procedure. In the current work, a polynomial regression, or in some cases, a linear regression was applied in order to generate a line of best fit. Typically a third order polynomial was sufficient to capture the trends of the performance plots, though in some instances fourth and fifth order polynomials were necessary. These regression equations were generated using the Microsoft Excel 2007 software package. Excel includes a trend line feature which accommodates polynomial and linear regression curve-fitting. 19

34 The stage stacking procedure is initiated by specifying the stage 1 inlet flow coefficient φ 1 and compressor corrected speed N c. Selection of these parameters sets the values of ψ, η, k b,1, and k b,2 for stage 1 according to the regression equations for the stage 1 performance curves. Stage 1 P 1 and T 1 must also be specified. In the current work, these inlet conditions were set to equal NASA standard sea-level values. Stage 1 inlet flow coefficient φ 1 is used to calculate the stage 1 inlet corrected axial velocity as given by Eq V z,1 T = φ 1 1 T sl U m,1 T 1 T sl Eq. 3.8 This corrected velocity is subsequently substituted into Eq. 3.6 to solve for stage 1 inlet corrected flow. The effective area term in Eq. 3.6 is defined in Chapter 4, where the stage 1 inlet blockage k b,1 is determined from the stage 1 inlet effective blockage curve regression equation. The stage 1 inlet midspan rotor speed U m,1 is given by Eq Next, the stage total pressure ratio is calculated by rearranging Eq. 3.1, as given by Eq P 2 = 1 + P 1 ψ U m,1 T 1 T sl c p T sl γ avg 2 γ avg 1 Eq. 3.9 The ideal work coefficient ψ is obtained from the stage 1 ideal work coefficient curve. The stage total temperature ratio is calculated by rearranging Eq. 3.4, as given by Eq

35 2 T 2 T 1 = 1 + ψ U m,1 T 1 T sl ηc p T sl Eq The efficiency η is obtained from the stage 1 adiabatic efficiency curve. The total pressure and total temperature ratios are used to calculate the stage 1 exit corrected flow given by Eq and corrected rotor speed given by Eq m T 2 T sl P 2 P sl = m T 1 T sl P 1 P sl T 2 T 1 P 2 P 1 Eq U m,2 T = 2 T sl U m,1 T 1 T sl r m,2 1 r m,1 T Eq T 1 The stage 1 exit corrected flow given on the left-hand-side of Eq is used with the isentropic relation given in Eq to calculate the stage 1 exit corrected axial velocity. m T 2 T sl A ef,2 P 2 P sl T 1 2 T sl = V z,2 V z,2 T 2 T sl 2 1 2c p T sl cos 2 α 2 1 γ 2 1 g c P sl RT sl Eq In the current work, the stage exit corrected axial velocity in Eq was iteratively solved using the Goal Seek function in the Microsoft Excel 2007 software package, which employs a bisection method. Here, α 2 is the design value of the stage 1 midspan exit absolute flow angle, and the effective area at the stage exit station A ef,2 is described in Chapter 4. The ratio of the stage 1 exit corrected axial velocity to the corrected rotor speed gives the stage 1 exit flow coefficient φ 2, as in Eq The stage 1 exit flow coefficient is 21

36 assumed to be equal to the stage 2 inlet flow coefficient. This assumption is valid because the stage 1 exit station and stage 2 inlet station are defined at the same axial location, as was illustrated in Figure 3.1 previously. The value of stage 2 inlet flow coefficient along with the corrected speed sets the values of ψ, η, k b,1, and k b,2 for stage 2 according to its performance curves. The procedure is repeated for stage 2, yielding the stage 2 total pressure ratio and total temperature ratio, as well as the stage 3 inlet flow coefficient. The process is repeated through the entire machine for each stage. Since most modern axial compressor designs, including the one studied in the current work, feature interstage bleeds, the stage stacking procedure outlined by Robbins and Dugan was updated to account for mass flow removal due to these bleeds. This reduction in mass flow rate affects the axial velocity, and thus, the flow coefficient. The mass flow rate downstream of a bleed is taken as the compressor inlet mass flow rate less the bleed flow rates from all upstream bleeds. This value of mass flow rate downstream of the bleed is used in Eq. 3.6 to calculate the inlet corrected axial velocity for the stage downstream of the bleed. An assumption is made here that total pressure and total temperature are constant across the bleed, which is satisfactory for the bleed rates chosen in this study. The values for bleed rate were based on the nominal values at the design point. Note that Eq. 3.6 requires a value for stage inlet effective area A ef,1 in order to solve for corrected axial velocity. This effective area is given by the effective blockage factor at 22

37 the stage inlet k b,1, which is a function of stage inlet flow coefficient φ 1, as will be shown in Chapter 4. Since this flow coefficient is not known a priori, an initial guess is set. This value, along with the stage blockage curve, establishes an initial value for A ef,1, so that stage inlet axial velocity can be calculated according to Eq Next φ 1 is calculated as in Eq The value of φ 1 is used to update A ef,1 with the aid of the stage blockage curve. This in turn updates the value of the inlet corrected axial velocity, which again updates φ 1. Thus, φ 1 for a stage immediately downstream of a bleed is iteratively updated until A ef,1 converges to several decimal places. In this way, a stage-by-stage calculation is carried out in order to find the performance of each stage at a given operating point. The overall compressor total pressure ratio and total temperature ratio are given by the product of the respective ratios across each stage. The stage stacking procedure described above includes an effective area and blockage parameter (detailed in Chapter 4), and a method to account for mass flow removal due to bleeds located between stages. It is necessary to construct the performance curves for each stage in order to use the current stage stacking procedure to estimate design and off-design performance of the compressor. 3.3 Constructing the Stage Performance Curves The stage stacking procedure defined above requires, among other information, the fully defined performance curves of every stage in the compressor. During preliminary design, the performance curves are estimated from empirical correlations derived from test data. These correlations are well grounded near design point. However, at off- 23

38 design conditions, their ability to estimate correct levels of performance is very often found to be lacking. In the current work, these stage characteristics were constructed using flow information extracted from CFD simulations generated using the APNASA CFD solver (Adamczyk, et al. 1990). This database of CFD simulations had to be sufficiently extensive to allow a stage stacking method to be used to estimate off-design performance during preliminary design. Simulations of the entire compressor will not, in general, yield the information required to fully populate the performance curves for all of its stages. This is because certain stages will stall or choke, depending on operating conditions, well before others due to the way the stages are aerodynamically matched. In order to fully define the near-stall and near-choke portions of the characteristics for each stage, it is necessary to simulate small groups of stages in isolation. Since stage characteristics were constructed from simulations of small groups of stages, it was necessary to show that stage characteristics constructed from these simulations matched those constructed from simulations of the entire compressor. Thus, a design speed simulation of stages 1 through 6 was generated and throttled along a speed line. Stage performance curves constructed from this simulation were compared to those constructed from a simulation of the entire compressor at design speed. The stage 1 performance curves of these two simulations are shown in Figure

39 Figure 3.3: Stage 1 characteristic from simulation of stages 1-6 compared to that from full compressor The stage 1 performance curves show normalized values of ideal work coefficient and adiabatic efficiency as a function of normalized stage inlet flow coefficient. The normalized ideal work coefficient is given by Eq. A.4. ψ = ψ ψ ref Eq Here, ψ* is the normalized ideal work coefficient, ψ is the ideal work coefficient as defined in Eq. 3.1, and ψ ref is the design point value of ideal work coefficient for the stage. Efficiency is normalized by the minimum and maximum values for this stage, as given by Eq. A.4. η = η η min η max η min Eq

40 Here, η* is the normalized adiabatic efficiency and η is the efficiency as defined in Eq The maximum value of efficiency η max is the peak efficiency for the stage, and the minimum value of efficiency η min is the efficiency associated with the simulated operating point yielding the lowest efficiency for the stage. All values of flow coefficient in the current work are normalized by a representative value, as given by Eq. A.4. φ 1 = φ 1 φ ref Eq Here, φ 1 * is the normalized stage inlet flow coefficient, φ 1 is the stage inlet flow coefficient as defined in Eq. 3.5, and φ ref is some reference value of flow coefficient. Points shown in blue are the results from the full compressor simulation and points shown in red are the results from the simulation of stages 1 through 6. The left-most points from both sets of data correspond to the near-stall points for the two simulations. Two major points are to be made from the results shown. Firstly, it is clear that the stage 1 characteristic constructed from the simulation of stages 1 through 6 lays on top of the stage 1 characteristic constructed from the full compressor simulation. Therefore, it is valid to use results from simulations of groups of stages in place of full compressor simulations for the purposes of generating these stage performance curves. Secondly, it is evident that the full compressor cannot adequately simulate near-stall behavior of stage 1. At design speed, the full compressor simulation goes into numerical stall at a much greater flow coefficient than the simulation of stages 1 through 6. This is because a stage downstream of stage 6 is thought to be the stall limiting stage at design speed. Exclusion of this stage in the simulation of stages 1 26

41 through 6 allows stage 1 to be throttled to much lower flow coefficients as compared to the full compressor case. This demonstrates the need to break the compressor into groups of stages in order to fully populate the stage characteristics. It was noted by Robbins and Dugan (1965) that the stage performance curves plotting ψ and η as a function of φ 1 collapse to a single line irrespective of wheel speed if the stage inlet relative Mach number is below approximately For stages with an inlet relative Mach number greater than 0.75, Robbins and Dugan remarked that the characteristics at different speeds do not collapse, and the plot is presented as a family of curves at different corrected speeds. For the compressor studied in the current work, the stage inlet relative Mach number falls below 0.75 for stages 5 and beyond. Therefore, it was hypothesized that the performance curves would collapse for stages 5 through 13, but would not collapse for stages 1 through 4. To test this hypothesis, simulations of stage 1 through stage 6 were generated at various corrected speeds so that their stage performance curves could be plotted and analyzed. Stages 1 through 6 of the compressor were simulated (called the front block), first at design speed, and then at part-speeds. The inlet boundary condition for these simulations was fixed to be identical to the inlet boundary condition of the full compressor simulation irrespective of operating conditions. The initial value for the exit boundary condition of hub static pressure was extracted from a full compressor simulation at design speed, at a location downstream of the stator 6 trailing edge. A 27

42 speed line of the front block was generated from near-choke to near-stall at design speed by systematically varying the exit boundary condition in steps of 5%-10% of its initial value. A 97% corrected speed simulation of these front stages was then generated using an initial flow field from a parent simulation which was a lightly loaded design speed front block simulation. The physical shaft speed and bleed rates were reduced to 97% of their design values. The exit boundary condition for the part-speed simulation was set at a 10% reduction of the parent simulation s exit boundary condition value. A speed line was then generated from near-choke to near-stall for the 97% speed simulation by systematically varying the exit hub static pressure in steps of 5%-10% of its initial value. A lightly loaded 97% speed simulation was used as a parent simulation to initialize a simulation at even lower corrected speed. This procedure was repeated to generate subsequent simulations at reduced speeds. Flow information from these front block simulations was extracted to calculate the performance parameters defined in Section 3.1. Plotting the performance curves for these stages across the various speeds confirmed the remarks of Robbins and Dugan; for those stages with inlet relative Mach number greater than 0.75, stage performance plots are presented as a family of curves for different speeds. This was the case for stages 1 through 4, shown in Figure 3.4 through Figure 3.7. For stage 5, the stage inlet relative Mach number dropped below 0.75 and the performance at different speeds collapsed to a single curve. The collapse of the performance curves is shown in Figure 3.8 and Figure

43 Increasing Corrected Speed Figure 3.4: Stage 1 performance curves at 6 different corrected speeds These figures show normalized values of ideal work coefficient and adiabatic efficiency as a function of inlet flow coefficient for each stage, with normalizations as described previously. The design speed simulations are shown as blue diamonds. The remaining sets of symbols show results for simulations at various corrected speeds. The stage 1 performance plot in Figure 3.4 shows discrete sets of curves at the different corrected speeds, especially at low flow coefficients. A black arrow indicates the direction of increasing corrected speed. The efficiency curves more clearly show the differences between speeds. 29

44 Increasing Corrected Speed Figure 3.5: Stage 2 performance curves at 6 different corrected speeds Figure 3.5 shows the stage 2 performance curves at various corrected speeds. As in Figure 3.4, it is observed that the performance curves vary according to the speed. 30

45 Figure 3.6: Stage 3 performance curves at 6 different corrected speeds The performance plot of stage 3 is shown in Figure 3.6. As in the plots of stages 1 and 2, there are distinct curves readily identified according to their corrected speed. 31

46 Figure 3.7: Stage 4 performance curves at 6 different corrected speeds By stage 4, the performance curves begin to collapse, as shown in Figure 3.7. This stage has inlet relative Mach numbers slightly greater than the threshold value of Figure 3.7 shows that stage 4 is beginning to choke in the simulations at Part-Speeds 4 and 5. This is apparent from the slope of the efficiency curve, which starts becoming vertical at high flow coefficients. Choking is also indicated by the low values of normalized ideal work coefficient at high flow coefficients. It was hypothesized that the choked stage at the part-speeds of interest in the current work would be downstream of stage 5. This hypothesis will later be shown to be correct in Section 5.2. Thus, it was unnecessary to fully construct the deep choke portions of the performance curves for stages 1 through 5. 32

47 Figure 3.8: Stage 5 performance curves at 4 different corrected speeds By stage 5, the performance curves at different speeds have essentially collapsed to a single curve. This is shown in Figure 3.8. There is some scatter of points, especially near peak efficiency, but it is difficult to identify distinct curves based solely on different corrected speeds. The inlet relative Mach number for this stage is well below the critical value of

48 Figure 3.9: Stage 6 performance curves at 4 different corrected speeds Finally, the performance plot of stage 6 in Figure 3.9 shows quite clearly that the curves for different corrected speeds have collapsed to a single line. The inlet relative Mach number for this stage is even lower than that of stage 5. The performance curves show that stage 6 is choked at the high end of its flow coefficient range, as indicated by the slope of the efficiency curve as well as the value of normalized ideal work coefficient approaching zero. Since downstream stages maintain relative Mach numbers below the threshold value of about 0.75, it was assumed that stages 7 through 13 would likewise exhibit a collapse of the curves irrespective of corrected speed. This assumption allows for the population of 34

49 performance curves for stages 7 through 13 using flow information extracted only from design speed simulations. These stages did not require extensive simulation at partspeeds since their performance curves would collapse to a single line irrespective of shaft speed. With the performance curves of stages 1 through 6 populated across various speeds, focus was turned to constructing the performance curves of stages 7 through 13 using design speed simulations. Populating the choke-side of the performance curves for these middle and rear stages was critical. This is because the stages towards the back of the machine are the ones expected to choke at low speeds, thereby establishing the limit of compressor operability at part-speed. The general method to obtain these simulations was to simulate a group of the middle or rear stages at design speed. Middle stages are stages 7 through 9, and rear stages are stages 10 through 13. The exit static pressure boundary condition for each group of stages was systematically reduced until the last stage in the group, which was the stage of interest, choked. Choking was characterized by a large drop in stage efficiency, the result of regions of separation along the pressure side of the rotor or stator, and/or the development of local supersonic flow within the stage. Once it was determined that the last stage in the simulation had choked, a new simulation was initialized which included an additional stage downstream of the choked stage. The exit static pressure boundary condition was once again systematically reduced until the last stage in this new simulation was choked. The details of this process follow. 35

50 A simulation of stages 8 through 10 was generated with an initializing flow field extracted from a parent simulation of the full compressor at design point. Flow information extracted from this parent simulation provided the inlet and exit boundary conditions. The design speed simulation of stages 8 through 10 was throttled down by incremental 5%-10% decreases in the exit static pressure boundary condition. The first two stages in this simulation, stages 8 and 9, acted as flow generators for the downstream blade rows. A flow generator of two stages was deemed sufficient to simulate the multistage environment, as shown in Figure Figure 3.10: Stage 10 inlet total pressure profile comparison 36

51 This figure compares the normalized stage 10 inlet total pressure profiles of a full compressor simulation to that of a rear block simulation which utilized stages 8 and 9 as flow generators. There is a good match between these two profiles, indicating that a two-stage flow generator upstream of the stage of interest (stage 10) is sufficient to produce the flow expected in the multistage environment. Performance parameters ψ, η, k b and φ were calculated for stage 10 after each change in throttle position. The backpressure was reduced until stage 10 was choked. A sample result of the stage 10 characteristic is provided in Figure Figure 3.11: Stage 10 performance curves choke-side defined 37

52 This shows the fully defined choke-side of the respective curve using flow information from simulations of stages 8 through 10. These points are shown as blue diamonds. The performance parameters extracted from simulations of the full compressor are plotted as grey circles. These grey points again serve to show the relatively small range of flow coefficient over which this stage operates within the full compressor. Because of the stage interactions, the points obtained by simulating stages 8 through 10 could not be obtained from a simulation of the whole compressor. It should be noted that the blade rows downstream of stage 10 were excluded from the simulation used to populate the choke-side of the stage 10 performance curves. In order to understand the effect of the exit boundary condition on the performance curve of stage 10, a new simulation of stages 8 through 11 was initialized using a flow field extracted from the parent design speed full compressor simulation. The inlet and exit boundary conditions were again set according to flow information extracted from the parent simulation. The exit static pressure boundary condition was systematically reduced until stage 11 choked. The performance parameters for stage 10 were generated using flow information from this new set of simulations. These new points are shown in Figure 3.12 as green triangles. 38

53 Figure 3.12: Stage 10 performance curves choke-side further defined It is clear from Figure 3.12 that the results from the two simulations, stages 8 through 10 and stages 8 through 11, lay on a single curve. The conclusion drawn from this figure was that the omission of stage 11 had negligible impact on the stage 10 performance curves. There was no need to model additional blade rows downstream of the stage of interest when throttling it to choked conditions. Simulations of stages 8 through 11, 8 through 12, and 8 through 13 were generated and throttled down to near-choke conditions in this manner. Flow information extracted from these CFD simulations was used to plot the stage performance curves for stages 10 39

54 through 13. Shown below in Figure 3.13 through Figure 3.16 are the fully defined performance curves of stages 10, 11, 12, and 13. Figure 3.13: Stage 10 performance curves 40

55 Figure 3.14: Stage 11 performance curves 41

56 Figure 3.15: Stage 12 performance curves 42

57 Figure 3.16: Stage 13 performance curves In order to populate the performance curves of stages 7 through 9, the procedure described above was repeated to generate simulations in which stages 5 and 6 acted as flow generators for the downstream blades. Again, the backpressure for these simulations was systematically reduced to choke the stages of interest, such that nearchoke simulations of stages 7, 8, and 9 were achieved. Flow information from these simulations was extracted to plot the stage performance curves for stages 7 through 9. The performance curves of stage 7, 8, and 9 are shown in Figure 3.17 through Figure

58 Figure 3.17: Stage 7 performance curves 44

59 Figure 3.18: Stage 8 performance curves 45

60 Figure 3.19: Stage 9 performance curves The performance curves of all stages were fully populated in the manner detailed above. Construction of these ψ-φ 1 and η-φ 1 curves was a requirement to use the stage stacking procedure described in Section 3.2. Recall that information on blockage at the stage inlet and exit stations was also a requirement of this stage stacking procedure. The development of the blockage characteristics for each stage is discussed further in the next chapter. 46

61 4 Blockage Generally, a blockage term is used to scale effective flow area in the continuity equation within mean-line and throughflow design codes (Casey and Robinson 2010). A generalized term for blockage within the continuity equation is developed in Appendix A. This blockage term should account for the effects of metal blockage due to the blades, as well as endwall blockage due to boundary layer growth and secondary flows such as tip leakage flow. Historically, blockage in the endwall regions has been modeled as a boundary layer displacement thickness along the inner and outer walls of the flow path, as shown in Figure 4.1. ROTOR Casing Boundary Layer Core Flow Hub Boundary Layer STATOR Figure 4.1: Illustration of boundary layer displacement thickness within a compressor stage Empirical correction factors are frequently used in order to match the blockage model to test data (Wright and Miller 1991). The current work has defined a blockage 47

62 parameter which mathematically closes the relations used within the one-dimensional stage stacking procedure described by Robbins and Dugan (1965). The blockage defined here is determined by using flow information extracted from CFD simulations of the compressor described in Section 3.3. Robbins and Dugan (1965) noted that the geometrical annulus area is commonly used in the equations of the stage stacking procedure, given previously by Eq. 3.6 and Eq. 3.13, but that more realistic values for corrected axial velocity are obtained when an effective flow area is used. This effective flow area is needed because the endwall regions in turbomachinery flows are viewed as boundary layers. There was no method detailed by Robins and Dugan to define this effective area term further. Therefore, a goal of the current work was to develop a new definition of effective flow area A ef in an effort to account for the blockage associated with endwall flows. The effective area was defined such that mathematical closure was achieved between the two equations given in Eq. 4.1 and Eq m T 1 T sl A ef,1 P 1 P sl = g cp sl T sl γ 1 p s,1 R M 1 cos α γ = 1 + γ 1 1 P 1 2 γ 1+1 M 2 2γ Eq. 4.1 γ 1 M 2 γ Eq. 4.2 Eq. 4.2 was rearranged to give Mach number M 1 as a function of the ratio of static to total pressure. This expression for Mach number was substituted into the right-hand- 48

63 side of Eq The equation was then simplified and rearranged to solve for the effective flow area. The result is given by Eq A ef,1 = m T 1 T P 1 sl P sl g c P sl γ 1 T R p γ 1 +1 s,1 2γ P 1 cos α 1 p s,1 sl 1 P γ 1 1 γ γ 1 1 Eq. 4.3 This expression was evaluated using mass averaged total conditions P 1 and T 1, and area averaged static pressure p s,1, which were extracted from CFD simulations. The design value of midspan absolute flow angle α was used. The mass flow rate ṁ was the stage inlet value, which accounted for flow removal due to bleeds. An effective blockage parameter k b was then defined by Eq k b = 1 A ef A an Eq. 4.4 Here, A an is the geometrical annulus area, and A ef is the effective flow area as defined in Eq The effective blockage parameter was calculated at the inlet and exit stations of each stage using flow information extracted from the CFD simulations described previously in Section 3.3. The effective blockage defined above was compared to the blockage parameter described by Khalid (1995). Khalid s definition of blockage is analogous to the definition of displacement thickness, as in Eq A b = 1 ρu m ρ e U e da Eq. 4.5 Khalid s definition of blockage was based on observable flow physics within cascade tests and CFD simulations. The integral in Eq. 4.5 is taken over a plane at the exit of the 49

64 blade row. Here, ρ e and U e refer to the density and velocity, respectively, at the edge of a defect region in the flow. The edge criterion of this defect region was based on (ρu m ), which is the gradient of momentum in the core flow direction. A cutoff value for the edge criterion is the minimum value not found in the core region, (Khalid et. al. 1999). Whereas A ef was developed to provide mathematical closure to the equations used within the stage stacking procedure, A b as defined by Khalid is based upon the idea of displacement thickness, which relates to the flow physics associated with the endwall boundary layer growth. Existing post-processing software which calculated this blockage using CFD simulations was used in the current work (Khalid 2000). Stagewise distributions compare the effective blockage defined in the current work, and Khalid s definition of blockage based on displacement thickness. Values of blockage are normalized by the stage 1 design point value. These stagewise distributions are compared at design point, at a near-stall point at 100% corrected speed, and at a part-speed near-stall point in Figure 4.2, Figure 4.3, and Figure 4.4 respectively. 50

65 Figure 4.2: Stagewise blockage distributions at design point 51

66 Figure 4.3: Stagewise blockage distributions at 100% speed near-stall point 52

67 Figure 4.4: Stagewise blockage distributions at a part-speed near-stall point These plots show that the effective blockage parameter agrees well with the blockage as defined by Khalid. This indicates that the two different calculations of blockage may be quantifying the same physical phenomenon; namely, the endwall boundary layer growth through the compressor. For both parameters, there are stages at the near-stall conditions which have markedly higher values of blockage. For the 100% speed case, stages 8 and beyond exhibit higher levels of blockage than do the upstream stages. It will be shown in Section 6.3 that stage 8 is likely a stall-limiting stage for this compressor at 100% speed, so this increase in blockage beginning at this stage may be indicative of its near-stall behavior. For the near-stall simulation at part-speed, stages 1 and 13 both have higher levels of blockage as compared to other stages. For stage 1, this may be 53

68 due to its near-stall behavior, as this is likely a stall-limiting stage at part-speed. However, stage 13 is highly choked in this simulation, and its high value of blockage is likely reflective of the separated flow associated with a choked stage, rather than any sort of near-stall behavior. For stages 1 through 4, this effective blockage parameter showed good correlation with stage inlet flow coefficient for a given speed. An example of the normalized stage 1 exit blockage plotted as a function of stage inlet flow coefficient for various corrected speeds is shown in Figure 4.5. Figure 4.5: Stage 1 exit effective blockage curves 54

69 Much like the ideal work coefficient and efficiency shown previously, the effective blockage curves across different corrected speeds collapsed to a single line for stages 5 and beyond. The stage 5 exit effective blockage curve shown in Figure 4.6 exemplifies this collapse irrespective of speed. Figure 4.6: Stage 5 exit effective blockage curve These blockage curves were incorporated into the stage stacking procedure, expanding upon the method described by Robbins and Dugan. The change in effective flow area as a result of the blockage term has a strong effect on axial velocity, and thus, the matching between stages. Hence blockage is an important parameter to specify in the current stage stacking procedure. 55

70 5 Applications of the Stage Stacking Procedure The primary utility of the stage stacking procedure is to generate estimates of overall compressor performance. Performance estimates at 97% corrected speed are compared to CFD results in Section 5.1. Another utility of the stage stacking procedure is the ability to explore the operability limits of the compressor. This is shown in Section 5.2, wherein an example is detailed which establishes the minimum corrected speed that allows stable operation of the compressor. Another utility is presented in Section 5.3, which presents a bleed sensitivity study highlighting the impact of stage 5 bleed rate on the part-speed operability range of the compressor. Finally, in Section 5.4 it is shown how the stage stacking procedure can be used to generate reasonable initial flow estimates for a complete compressor simulation at part-speed operation. 5.1 Performance Estimates at 97% Corrected Speed The stage stacking procedure was used to generate several performance estimates associated with various stage 1 inlet flow coefficients for 97% corrected speed. These stage stacking estimates are shown for stage 1 in Figure

71 Figure 5.1: Stage stacking estimates along the stage 1 performance curve, 97% speed The stage stacking calculation was performed for each of the blue points in Figure 5.1. The overall compressor performance was then calculated, and compared to CFD results of the entire compressor at 97% corrected speed. The results are shown in Figure

72 Figure 5.2: Stage stacking performance estimate compared to CFD at 97% speed The charts in Figure 5.2 show overall total pressure ratio and adiabatic efficiency as a function of compressor inlet corrected flow. The total pressure ratio and inlet corrected flow are normalized by their design values, and the efficiency is normalized by the minimum and maximum values. The stage stacking procedure estimated the overall total pressure ratio to within 1.6% of the CFD result for approximately equal inlet corrected flow. The efficiency estimated by the stage stacking calculation was within 0.3 points of the CFD result for approximately equal inlet corrected flow. There was no a priori expectation as to how accurately the stage stacking calculation would estimate the overall performance of the compressor. The adequacy of the accuracy of this stage stacking procedure is largely dependent upon the application of the results. For 58

73 example, this stage stacking procedure may be sufficiently accurate for preliminary design work, yet may fall short if used for detailed final design analyses. 5.2 Estimation of the Minimum Corrected Speed The stage stacking procedure was used to estimate the minimum corrected speed that allowed stable compressor operation with geometry fixed. This was done by performing the stage-by-stage calculation at various stage 1 inlet flow coefficients and corrected speeds. The minimum speed which allowed stable operation of the compressor is referred to as part-speed N%. It will be shown that a stage stacking estimate at a speed 2% lower than N% speed, referred to henceforth as N2% speed, indicated that the compressor would not operate in a stable fashion. 59

74 Figure 5.3: Stage 1 performance curves and stage stacking estimate at N2% speed Figure 5.3 shows the stage 1 N2% performance points as red squares. The blue diamond depicts the stage stacking estimated performance for the stage. The minimum value of flow coefficient was specified for the stage 1 inlet flow coefficient for this stage stacking example. Selection of this flow coefficient determined the work coefficient and adiabatic efficiency according to the polynomial best-fit curves. A stage-by-stage calculation was performed as described in Section

75 Figure 5.4: Stage 9 performance curves and stage stacking estimate at N2% speed As the stage-by-stage calculation progressed through the machine, it was noted that the stage stacking estimate of stage 9 performance indicated that this stage was nearly choked, as shown in Figure 5.4. This is readily seen on the efficiency plot, which shows that the stage stacking point is nearly on the vertical portion of the curve. The choked operation of stage 9 confirms the hypothesis alluded to in Section 3.3, which estimated that a stage downstream of stage 5 would be the choked stage at the part-speed operating points of interest. Attempting to continue the stage stacking procedure beyond stage 9 proved to be futile. This was because the stage 10 inlet flow coefficient estimated by the stage stacking procedure was greater than the maximum possible value attainable by the stage, as shown in Figure

76 Figure 5.5: Stage 10 performance curves and stage stacking estimate at N2% speed The stage 10 inlet flow coefficient estimated via stage stacking was far greater than the maximum, or choking, flow coefficient. Thus, this estimated flow coefficient could not be used to establish values for ideal work coefficient or efficiency, and thus the stage stacking routine was terminated. This result indicated that the compressor could not operate at the given inlet conditions. In order to achieve operability at part-speed N2%, the compressor inlet flow coefficient would have to decrease. However, near-stall value of stage 1 inlet flow coefficient was specified in this case. The compressor would likely stall if it were to operate at a lower inlet flow coefficient. Hence at part-speed N2%, there is no stable region of operation for this compressor, given nominal bleed settings and IGV/VGV settings. 62

77 A separate stage stacking estimate was generated for part-speed N%, which is a corrected speed 2% greater than part-speed N2%. A near-stall value of stage 1 inlet flow coefficient was picked along the N% speed stage 1 performance curves, as shown in Figure 5.6. Figure 5.6: Stage 1 performance curves and stage stacking estimate at N% speed The stage stacking procedure was carried out through the compressor. The resulting stage 13 performance is shown in Figure

78 Figure 5.7: Stage 13 performance curves and stage stacking estimate at N% speed The stage stacking result at stage 13 indicated that this stage was close to choking. Since this is the last stage, there is no issue with the flow coefficient of any subsequent stage downstream being greater than its value at choke. These results indicated that the compressor will operate at part-speed N%. Thus, the minimum speed for stable operation with nominal IGV/VGV settings and bleed rates is between speeds N% and N2%. As will be detailed in Section 5.4, a full compressor CFD simulation was executed at N% corrected speed which converged to an overall total pressure ratio and inlet mass flow rate within 1.3% and 0.3%, respectively, of the stage stacking estimate. The inlet boundary conditions of this simulation were consistent with those used in the stage stacking procedure. 64

79 5.3 Operability and Bleed Sensitivity Estimates A study was performed to assess the impact of the bleed rates on the operability range of the compressor using the stage stacking procedure. It was demonstrated in Section 5.2 that the compressor would not operate in a stable way at corrected speed N2%. Recall that this was due to an estimate for stage 10 inlet flow coefficient which was greater than the value at choke. To achieve operability, the flow coefficient of stage 10 was reduced by extracting some amount of mass flow entering stage 10. This was accomplished by increasing the stage 5 bleed rate in a bleed sensitivity study detailed herein. The bleed sensitivity study was conducted to estimate the stage 5 bleed rate required such that only stage 13 operated near-choke, and all other stages operated within their stable operating ranges. This was accomplished by iteratively varying stage 5 bleed rate within the stage stacking procedure and performing the stage-by-stage calculation with the stage 1 inlet flow coefficient fixed to its near-stall value for N2% speed. These iterations were performed until the stage stacking procedure estimated a near-choke flow coefficient for stage 13, with all upstream stages operating in their stable operating ranges. The resulting stage 5 bleed rate was approximately 7 times greater than the nominal rate. Values of static pressure resulting from the stage stacking estimate were used to generate a CFD simulation at N2% speed, as described in Section 5.4. The stage 5 bleed 65

80 rate was increased to the new value in this simulation. Stage performance parameters calculated from this CFD simulation were then compared to the stage stacking estimate. Figure 5.8 shows the stage 1 performance curves. Figure 5.8: Stage 1, N2% speed stage stacking estimate and CFD (increased bleed rate) This plot shows that the CFD simulation, shown as a green triangle, converged to a slightly lower stage 1 inlet flow coefficient at N2% speed. The result of increasing the stage 5 bleed rate is shown in Figure

81 Figure 5.9: Stage 9, N2% speed stage stacking estimate and CFD (increased bleed rate) Here, the stage stacking estimate of stage 9 inlet flow coefficient with nominal stage 5 bleed rate (shown as an orange circle) is compared to that with increased stage 5 bleed rate (shown as a blue diamond). The increase in stage 5 bleed rate resulted in a significantly reduced estimate of stage 9 inlet flow coefficient. Examining stage 13 performance curves in Figure 5.10 shows that the CFD simulation resulted in stage 13 being choked. This was the goal of increasing the stage 5 bleed rate in the stage stacking exercise, as discussed above. 67

82 Figure 5.10: Stage 13, N2% speed stage stacking estimate and CFD (increased bleed rate) Thus, the stage stacking procedure was able to describe a bleed schedule which could extend the operating range of this compressor down to part-speed N2%, which is a corrected speed 2% lower than N% speed. Increasing the stage 5 bleed rate in the compressor acted to reduce the flow coefficient for choked stages within the middle of the machine. The value for this bleed rate was determined by iterative calculations of the stage stacking procedure. The stage stacking calculation estimated the compressor inlet corrected flow to within 0.3% of the resulting CFD simulation. Overall total pressure ratio was estimated to within 1.3% of the CFD result, and overall adiabatic efficiency was estimated to within 0.42 points of the CFD result. 68

83 As a final step in this operability study, the stage 5 bleed rate in the CFD simulation was reduced from 7 times the nominal value back down to the nominal value. This resulted in numerical stall in the front stages of the compressor. This result confirmed the stage stacking estimate that the compressor could not operate at N2% speed. 5.4 Construction of Initial Guesses for Full Compressor Simulations Prior to this work, obtaining part-speed APNASA simulations of the full compressor proved to be difficult. The former approach to generate part-speed simulations was to decrease the backpressure while simultaneously reducing the shaft speed, starting with an initial flow field from a parent design speed simulation. The reduction in backpressure was achieved by reducing the hub static pressure at the compressor exit. The value of the hub static pressure was generally guessed as some percentage of that of the parent simulation. The change of the exit hub static pressure resulted in a transient fluctuation in flow conditions at the exit of the compressor. This transience propagated upstream with successive numerical iterations. Due to the large number of blade rows in the simulation, many hours of computing time were necessary to resolve the effects of the altered exit boundary condition at the front end of the compressor. Conversely, the reduction in shaft speed was experienced by the entire compressor simultaneously. Therefore, this approach caused the front stages of the compressor to operate at rather high blade loadings during part-speed operation. A simplified definition of ideal work coefficient in Eq. 5.1 serves to illustrate the complications of this approach. 69

84 ψ = H ideal U m 2 Eq. 5.1 Here, ΔH ideal is the ideal total enthalpy rise across a stage and U m is the midspan rotor speed. Consider the loading of the first rotor in the machine immediately after the shaft speed and backpressure are reduced from design point values down to values associated with a part-speed operating point. The reduced shaft speed is manifested in Eq. 5.1 as a reduction in U m. Since the reduction in backpressure was made far downstream of the first rotor, the flow through this rotor does not immediately feel the change in the compressor exit boundary condition. Thus, the total enthalpy rise across this first rotor is unchanged for many numerical iterations. Thus, the numerator of Eq. 5.1 is unchanged, while the denominator is reduced. The loading of this rotor becomes quite high, and remains high until the transient wave from the change in backpressure reaches the front of the compressor. Because of the large number of blade rows and axial nodes in the mesh, the front blade rows go into a numerical stall before this transient wave ever reaches the front stages, preventing a full compressor simulation from converging at part-speed operating points. A new approach to generate part-speed simulations of the full compressor was developed in the current work such that the occurrence of numerical stall was mitigated. Rather than attempting to bring the entire compressor down in speed at once, a new strategy was developed which involved the disassembly of the full compressor simulation into three groups of stages, or blocks, which were simulated separately. These block simulations were stepped down in shaft speed and loading one 70

85 by one and run until mass flow rate and total pressure converged. The flow fields of these block simulations were then reassembled and used to initialize a new full compressor simulation, which was capable of converging without any numerical stall issues at a part-speed operating point. The decision to simulate blocks of stages was motivated by a need to reduce the amount of iterations necessary for the front stages to experience the effects of the change to the exit boundary condition. This new approach can be thought of as effectively setting intermediate boundary conditions throughout the compressor so that a reduction in blade loading is felt more concurrently by all blade rows. The three block simulations were defined according to the locations of the three casing bleed locations in the compressor, located downstream of stages 5, 8, and 11. Therefore, the front block was comprised of the IGV through stator 6, the middle block was comprised of rotor 5 through stator 9, and the rear block was comprised of rotor 8 through the OGV. The overlapping of blade rows between these blocks was intentional, since the first several blade rows of the middle and rear block simulations acted as flow generators which established the multistage flow for the downstream blades, as previously illustrated by Figure The full compressor was reassembled using the flow fields of the IGV through stator 6 from the front block, rotor 7 through stator 9 of the middle block, and rotor 10 through OGV of the rear block. Note that the flow fields of the flow generating stages for the middle and rear block were excluded when reassembling the full compressor simulation. 71

86 The inlet boundary condition for the front block was identical to that of the full compressor simulation at design speed. The inlet boundary condition for the middle block was extracted from the front block simulation once it had converged, at a location upstream of the rotor 5 leading edge. Likewise, the inlet boundary condition for the rear block was extracted from the converged middle block simulation, at a location upstream of the rotor 8 leading edge. These inlet boundary conditions were set as radial profiles of total pressure, total temperature, tangential flow angle, and radial flow angle. The exit boundary conditions for the block simulations were prescribed values of hub static pressure. The values for the exit boundary conditions needed to be set such that they reflected the performance of the entire compressor at a stable part-speed operating point. The front block terminated with stator 6, the middle block terminated with stator 9, and the rear block terminated with the OGV. Therefore, there was a need to estimate the static pressures at these locations for a given shaft speed. These estimates were performed using the stage stacking procedure described previously. The stage exit total pressure, corrected flow, and effective area, which were obtained by the stage stacking procedure, were used to calculate the static pressure at each stage exit station according to the relation given by Eq

87 m T 2 T sl A ef,2 P 2 P sl = g cp sl γ γ T sl R p 2 2γ 2 cos α P 2 p γ 2 2 γ P 2 2 γ 2 1 Eq. 5.2 Here, a bisection method was used to iteratively solve for the static pressure p s,2. The estimated stage 6, stage 9, and stage 13 exit static pressures were used as initial guesses for exit boundary conditions of front, middle, and rear block CFD simulations at N% speed. Recall from Section 5.2 that this speed was estimated as being the minimum speed for stable compressor operation for fixed compressor geometry. Once the front, middle, and rear block simulations were converged, their flow fields were reassembled into a full compressor simulation at N% speed. The exit static pressure was set to be identical to the exit static pressure derived from the stage stacking procedure. This process mitigated the problems of numerical stall which were associated with attempts to generate part-speed APNASA simulations of this compressor prior to this work. The resulting full compressor simulation converged to a total pressure ratio and inlet mass flow rate to within 1.3% and 0.3%, respectively, of the stage stacking estimate. This CFD result was shown previously by green triangles in the figures of Section

88 6 Assessment of a Correlation for Stalling Stage Pressure Rise A semi-empirical correlation developed by C. C. Koch (1981) allowed for the estimation of the maximum static pressure rise of an axial flow compressor stage as a function of stage geometry parameters. The correlation contained corrections which accounted for effects of varying Reynolds number, rotor tip clearance, axial spacing, and velocity diagrams. The resulting stage static pressure rise coefficient was approximated well by the two-dimensional diffuser correlation of G. Sovran and E. D. Klomp (1967) at 9% inlet blockage. This correlation was shown by Koch to have credible results for several geometries; cascades, low speed rigs, and high speed compressors. The compressor designs that Koch studied were for aircraft turbine engine applications, and all designs had shrouded stators. An attempt was made to apply Koch s correlation with the compressor studied in the current work. Recall that this compressor is part of an industrial gas turbine engine; thus its stage inlet Reynolds numbers are 1 to 2 orders of magnitude greater than the compressors Koch studied in Additionally, the current compressor has cantilevered stators for stages 4 through 13, which presents a departure from the shrouded stator compressors Koch studied. A modified version of Koch s correlation was applied using flow information extracted from the CFD simulations generated in the current work. The results of this correlation were compared against the flow physics observed in the CFD simulations. 74

89 6.1 Background of the Correlation Koch (1981) attempted to draw an analogy between a two-dimensional diffuser and a compressor stage, both of which work by diffusing the fluid. Two-dimensional diffusers have a maximum static pressure coefficient that is determined by geometry parameters. This diffuser static pressure rise coefficient C p is defined in Eq C p = p s 1 2 ρu2 Eq. 6.1 Here, Δp s is the static pressure rise across the diffuser, ρ is the fluid density, and u is the mass-averaged velocity. The two-dimensional diffuser has a maximum static pressure coefficient at a single area ratio (A 2 /A 1 ) for a given diffuser-length-to-inlet-width ratio (N/W 1 ). This is shown in Figure 6.1 (adapted from Johnson 1998), which is a typical performance chart for two-dimensional diffusers, based on the data of Reneau, Johnston, and Kline (1964). Figure 6.1: Typical diffuser performance chart, adapted from Johnson (1998) 75

90 It is shown in Figure 6.1 that a maximum C p exists for a given value of N/W 1, corresponding to a single critical value of A 2 /A 1. Any further increase in area ratio beyond this critical value reduces the pressure recovery due to the increase in separated flow in the diffuser. Koch reasoned that since compressor stages also work by diffusing the fluid, the maximum static pressure rise across a compressor stage may similarly be determined by cascade geometry parameters, which are shown in Figure 6.2. Figure 6.2: Compressor blade geometry, adapted from Johnson (1998) Koch selected the arc length of the cambered airfoil L as the analogy to diffuser length N, where L is given in Eq L = 2π C 360 Φ 2 sin Φ Eq Here, Φ is the midspan camber angle in deg and C is the midspan chord length. Next, an analogy for diffuser inlet width W 1 was sought. This inlet width is held fixed when 76

91 diffuser data is correlated (Johnson 1998). Koch selected the blade row exit staggered spacing g 2 as the appropriate analogy to diffuser inlet width W 1. The exit value was chosen because it is roughly constant across the operating range, unlike the inlet staggered spacing g 1 which decreases as the compressor is throttled to higher backpressure. This is because staggered spacing g is a function of the pitch S and the relative flow angle β, as shown in Eq The blade row inlet flow angle increases as the compressor is throttled, whereas the blade row exit flow angle is roughly constant across the operating range. g = S cos β Eq. 6.3 Thus, Koch determined that the appropriate analogy to a diffuser s non-dimensional length N/W 1 was the cascade non-dimensional length L/g 2, shown in Eq L g 2 = 2π C 360 Φ 2 sin Φ 2 1 S cos β 2 Eq. 6.4 Values of L/g 2 for the rotor and stator of a given stage were calculated individually, and then combined into a weighted average stage value where blade row inlet dynamic head was the weight factor, as in Eq L L 2 V g 1 + L V 2 g 2 1 rotor 2 stator = Eq. 6.5 g 2 2 stage V 1 rotor + V 2 1 stator Here, V is the midspan relative velocity and V is the midspan absolute flow velocity. A static pressure rise coefficient C h based on ideal enthalpy rise across the stage was defined by Koch, as shown in Eq

92 C h = c p t 1 p s,2 p s,1 γ avg 1 γ avg 1 2 stage 2 V 1 + V rotor 1 stator 2 U 2 m,2 U 2 m,1 rotor 2 Eq. 6.6 In the current work, p was taken to be the area averaged static pressure, and static temperature t was given by Eq t 1 = T 1 p γ 1 1 s,1 γ 1 Eq. 6.7 P 1 Here P and T are mass averaged total pressure and total temperature, respectively, and U m is the rotor speed. We see that C h is composed of an ideal enthalpy rise across the stage (the first term in the numerator of Eq. 6.6), less any enthalpy change due to changes in midspan radius across the rotor (the second term in the numerator of Eq. 6.6), normalized by the sum of rotor and stator midspan relative dynamic heads (the denominator of Eq. 6.6). Koch outlined four corrections to this static pressure rise coefficient, which attempt to account for the effects due to varying Reynolds number, rotor tip clearance, axial spacing, and velocity triangles. Figure 6.3 shows the results of the correlation after making the aforementioned corrections. The curve in this figure is the two-dimensional diffuser correlation of Sovran and Klomp (1967) with 9% inlet blocked area. This curve defines the effective diffusion limit, with stalling static pressure coefficients for the majority of stages studied by Koch falling on or just below the curve. 78

93 Figure 6.3: Correlation for stalling static pressure rise coefficient, adapted from Koch (1981) Koch plotted the experimental values of stalling static pressure rise coefficient for stages of various geometries as a function of stage inlet Reynolds number, based on midspan chord length and velocity. These stalling values of pressure coefficient were calculated for stages with Reynolds numbers ranging from about 15,000 to 800,000. These coefficients were normalized by the value expected for a stage with Reynolds number of 130,000. A plot showing the effect of Reynolds number was presented by Koch, and the trendline of Koch is reproduced here in Figure

94 1.05 Normalized Stalling Static Pressure Rise Coefficient, C h /(C h at Re=130,000) Reynolds Number x 10-5 Figure 6.4: Effect of Reynolds number on stalling static pressure rise coefficient The curve in Figure 6.4 was used by Koch to adjust all pressure coefficients to the value expected at a Reynolds number at 130,000. This would be accomplished by dividing the value of C h given in Eq. 6.6 by the adjustment factor read from the curve in Figure 6.4, which is a function of the stage inlet Reynolds number. However, this adjustment was neglected in the current work. This is because the stage inlet Reynolds numbers for the industrial compressor studied in the current work are on the order of 2,000,000 to 4,000,000, which is an order of magnitude greater than any aero compressor that Koch analyzed. Furthermore, it can be seen that the largest impact on the stalling static pressure coefficient is for low Reynolds numbers stages. Thus, the correction for Reynolds number was neglected in the treatment of the compressor studied in the current work. 80

95 Koch showed the effect of rotor tip clearance on the stalling pressure rise coefficient. The stalling pressure rise coefficient for stages with varying rotor tip clearance and aspect ratio was normalized by the value expected for a normalized tip clearance of 5.5% of midspan staggered spacing. A trend line following Koch s data is reproduced in Figure Normalized Stalling Static Pressure Rise Coefficient, C h /(C h at ε/g 2 =0.055) Tip Clearance: Midspan Exit Staggered Spacing, ε/g 2 Figure 6.5: Effect of tip clearance on stalling pressure rise coefficient This curve was used in Koch s work to adjust the static pressure rise coefficient to the value expected for a normalized rotor tip clearance of 5.5% exit staggered spacing. This would be accomplished by dividing the value of C h, which had previously been adjusted for Reynolds number, by the adjustment factor read from the curve in Figure 6.5. It should be noted that only rotor tip clearance was addressed by Koch, as all compressor 81

96 stages he studied were of shrouded stator design. For the compressor which is the focus of the current work, stators 4 through 13 are cantilevered, and consequently have a hub clearance. To account for the effect of the leakage vortex associated with stator hub clearance, the methodology detailed by Koch was modified in the current work. Using the curve in Figure 6.5, the static pressure rise coefficient was adjusted to the value expected for a ratio of tip clearance to trailing edge staggered spacing of 5.5% for both the rotor and the stator individually, based on their respective values of tip clearance and trailing edge staggered spacing. These adjusted coefficients were then combined into a weighted average stage value as shown in Eq. 6.8 and Eq C h at ε g 2 = stage = Λ C h at ε g 2 = rotor Eq (1 Λ) f ε stator ε rotor C h at ε g 2 = stator Λ = p s,rotor p s,stage Eq. 6.9 Here, Λ is the degree of reaction based on static pressure rise. Note the factor f which is multiplied to the last term in the right-hand-side of Eq This factor is some function of the ratio of stator hub clearance to rotor tip clearance. This function f is defined such that f(1) = 1 and f(0) = 0. Thus, for compressor stages with shrouded stators, i.e. ε stator =0, this correction for tip clearances reduces to the one given by Koch. For stages with cantilevered stators, Koch s original correction for tip clearance is modified to account for stator hub clearance. For the current compressor, rotor and stator 82

97 clearances are taken to be of approximately equal magnitude, and the assumption is made that f=1 for stages with cantilevered stators. Thus, Eq. 6.8 simplifies to Eq. A.4 C h at ε g 2 = stage = Λ C h at ε g 2 = rotor Eq (1 Λ) C h at ε g 2 = stator The decision to use degree of reaction as the weight factor in Eq. A.4 is best explained by example. Consider a stage in which most of the diffusion occurs in the rotor. It can be argued that the tip leakage flow of this rotor should have a greater impact on the pressure rise capability of the stage as compared to the tip leakage flow of the stator. Thus, the adjustment for rotor tip clearance should carry greater weight than the adjustment for stator tip clearance. Since this stage corresponds to a high degree of reaction, it was postulated that Λ would be a suitable weight factor. This weight factor gives heavier influence to the tip clearance adjustment associated with the blade row that is responsible for the larger percent of diffusion. If the rotor and stator contribute equally to the static pressure rise across the stage, i.e. Λ=0.5, then the adjustment factors for both rotor and stator tip clearance are given equal weight. The third correction to the static pressure rise coefficient detailed by Koch accounted for the effect of axial spacing between the rotor and stator. Koch showed that stages of varying aspect ratio collapsed to the curve in Figure 6.6, showing the stalling static pressure rise coefficient as a function of axial spacing normalized by rotor pitch Δz/s. 83

98 The stalling static pressure rise plotted here was normalized by the value expected for a stage with Δz/s equal to Normalized Stalling Static Pressure Rise Coefficient, C h /(C h at Δz/s=0.38) Axial Spacing: Rotor Pitch, Δz/s Figure 6.6: Effect of axial spacing on stalling pressure rise coefficient Koch adjusted the static pressure rise coefficient by dividing it by the adjustment factor on the y-axis of Figure 6.6. The appropriate value was read off of the curve shown. This adjustment was made in the treatment of the compressor studied in the current work. The value for axial spacing between the rotor and stator and the value for rotor pitch were evaluated at midspan in order to apply this adjustment. A final adjustment to the static pressure rise coefficient was described in Koch s original work. This adjustment attempted to account for the effect of varying velocity diagrams due to differences in blade stagger and camber angles. Koch s procedure for applying 84

99 this adjustment began with defining a minimum possible blade row inlet velocity V min, which was given as a function of the blade row inlet relative and absolute flow angles β 1 and α 1 respectively. This minimum velocity was defined according to Eq V min = V 1 sin(β 1 + α 1 ) if (β 1 + α 1 ) 90 deg V 1 if (β 1 + α 1 ) > 90 deg Eq Koch used this minimum velocity to define a term which he called the effective dynamic pressure factor F ef, defined as the effective dynamic head divided by the midspan free stream dynamic head. This pressure factor was defined as a weighted average of the free stream dynamic head, the minimum possible velocity, and the rotor speed. The weight factors for this average were determined by Koch via trial and error to match his data, and are shown in Eq F ef = V ef 2 2 V = V V 2 2 min + 0.5U m,1 Eq V 1 Koch s final adjustment to the static pressure rise coefficient was to replace the rotor and stator free stream dynamic heads in Eq. 6.6 by their respective effective dynamic heads V ef. The rationale behind this adjustment was seemingly related to the recovery ratio R R, defined in Eq (Cumpsty 1989). R R = 1 cos(β 1 + α 1 ) cos(β 2 + α 2 ) cos α 1 cos β 2 cos β 1 cos α 2 Eq This recovery ratio is a measure of the ability of a blade row to eliminate any upstream disturbance in total pressure. If the upstream disturbance is completely eliminated downstream of the blade, R R is equal to 1. This is obtained when either (β 1 +α 1 ) or 85

100 (β 2 +α 2 ) is equal to 90 deg. If R R is much greater than 1, regions of high loss in [the upstream] frame of reference become regions of excess [total] enthalpy and pressure in the frame of reference downstream, (Cumpsty 1989). Essentially, this indicates that a blade row with R R greater than 1 acts to re-energize the endwall boundary layer flow from the upstream blade row. This is the case for blade rows in which either (β 1 +α 1 ) or (β 2 +α 2 ) is greater than 90 deg. Koch postulated that these types of blade rows might be less likely to stall. Just the opposite is true for blade rows for which R R is less than 1, corresponding to values of (β 1 +α 1 ) or (β 2 +α 2 ) which are less than 90 deg. For the compressor studied in the current work, values of (β 1 +α 1 ) or (β 2 +α 2 ) were generally found to be approximately 90±10 deg, indicating a recovery ratio near 1. Therefore, if the idea of recovery ratio was in fact the motivation behind Koch s fourth correction, it was deemed unnecessary to apply the correction for this compressor. 6.2 Results of the Correlation The static pressure rise coefficient was calculated as detailed in Section 6.1 using flow information extracted from full compressor simulations. Three particular simulations are of greatest interest; a simulation near the aerodynamic design point, a 100% speed simulation near the stall point, and a part-speed N% simulation near the stall point. The results from the aerodynamic design point simulation serve as a basis of comparison against the two near-stall simulations. 86

101 Figure 6.7 below plots the static pressure rise coefficients for stages in a design point APNASA CFD simulation after applying the adjustments for tip clearance and axial spacing as detailed previously. The two-dimensional diffuser correlation of Sovran and Klomp at 9% blockage is overlaid on this plot to establish the diffusion limit. This correlation is approximated over the domain 0 L/g by the polynomial curve defined in Eq C h = L L L Eq g 2 g 2 g 2 Aerodynamic Design Point Static Pressure Rise Coefficient, C h Non-Dimensional Diffusion Length, L/g 2 2D Correlation Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage 9 Stage 10 Stage 11 Stage 12 Stage 13 Figure 6.7: Correlation of static pressure rise coefficients for stages near design point Here we can see that there is an assortment of relative loading between the stages at the design point. Certain stages, such as stage 5, are quite close to the diffusion limit curve, whereas other stages, such as stages 12 and 13, are quite far from the estimated maximum diffusion point. It is of interest to examine how the static pressure rise 87

102 coefficients for these stages change as the machine approaches stall. Thus, similar plots were generated for the near-stall operating points at 100% corrected speed and at partspeed N%, as shown in Figure 6.8 and Figure % Speed, Near-Stall Point Static Pressure Rise Coefficient, C h Non-Dimensional Diffusion Length, L/g 2 2D Correlation Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage 9 Stage 10 Stage 11 Stage 12 Stage 13 Figure 6.8: Correlation of static pressure rise coefficients for near-stall point at 100% speed 88

103 Static Pressure Rise Coefficient, C h N% Speed, Near-Stall Point Non-Dimensional Diffusion Length, L/g 2 2D Correlation Stage 1 Stage 2 Stage 3 Stage 4 Stage 5 Stage 6 Stage 7 Stage 8 Stage 9 Stage 10 Stage 11 Stage 12 Stage 13 Figure 6.9: Correlation of static pressure rise coefficients for near-stall point at N% speed As shown in Figure 6.8, stages 5 and 8 nearly lay on the diffusion limit curve, indicating that these may be the stall-limiting stages at 100% corrected speed. Similarly, Figure 6.9 indicates that stages 1 or 2 may be the stall-limiting stages at part-speed N%, due to their proximity to the diffusion limit curve. In order to achieve a better understanding of how the static pressure rise coefficient is changing with operating conditions, a stagewise distribution is plotted. Figure 6.10 plots the stalling pressure rise ratio against the stage number. This ratio is taken as the stage static pressure rise coefficient divided by the value predicted by the two-dimensional diffuser correlation for that stage. Values of this ratio that approach 1 indicate that a stage is nearing its estimated diffusion limit, and may be nearing its stalling point. 89

104 Stagewise Distribution of Stalling Pressure Rise Ratio Aerodynamic Design Point 100% Speed, Near Stall Stalling Pressure Rise Ratio Stage Number Figure 6.10: Stalling pressure rise ratio - design point compared to 100% speed near-stall point Figure 6.10 indicates that every stage has moved towards its predicted stalling pressure rise. Furthermore, it can be seen that stages 5 and 8 approach stalling pressure rise ratios of 1, indicating that they are at their maximum predicted pressure rise. It should be noted that the reason for the large drop in stalling pressure rise ratio seen between stages 5 and 6 is not completely understood. It was conjectured that this may be a result of the casing bleed between stator 5 and rotor 6, which is unaccounted for within the correlation for static pressure rise coefficient. Irrespective of this anomaly, Figure 6.10 indicates that this compressor is well-designed, as it is favorable to establish the 90

105 middle stages as the stall-limiting ones at design speed. The front stages and rear stages have high stall margin at design speed. Stall margin in the front stages allows for greater part-speed operability range, as these are the stages which one would expect to stall at corrected speeds less than design. Likewise, stall margin for the rear stages allows for greater operability at over-speed operation. 1 Stagewise Distribution of Stalling Pressure Rise Ratio N% Speed, Near Stall Aerodynamic Design Point 0.8 Stalling Pressure Rise Ratio Stage Number Figure 6.11: Stalling pressure rise ratio - design point compared to N% speed near-stall point Figure 6.11 compares the stagewise distributions of stalling pressure rise ratio for the design point and the N% speed near-stall point simulations. Stages 4 and 5 have negligible changes in stalling pressure rise ratio between the two simulations. As the 91

106 corrected sped is decreased from design speed to N% speed, stages 1 through 3 are approaching the stalling pressure rise. Specifically, stages 1 and 2 show rather large increases in stalling pressure rise ratio, nearing their respective maximum pressure rise values. This indicates that either stage 1 or 2 is the stall-limiting stage at part-speed N%. Stalling of the front stages is what one may expect for a compressor operating at low corrected speed. Results of stages 6 through 13 at N% speed show large decreases in stalling pressure rise ratio as compared to the design speed results. The result for stage 13 shows that this last stage is completely choked and is in fact has a negative static pressure rise. Again, this result is to be expected, as choking of the rear stages is what one may anticipate for a compressor operating at low corrected speeds. 6.3 Connecting the Correlation Results to Flow Physics Captured by CFD Koch s correlation for maximum stage pressure rise was based on an analogy to twodimensional diffusers. In two-dimensional diffusers, the maximum pressure rise is dictated by flow separation in the endwall regions. It was thought that Koch s correlation would thus be an indicator of wall-stall in compressor stages, as opposed to blade-stall. The work of H. D. Vo (2001) indicated two critical flow features associated with spike-type stall inception. The first critical feature was the movement towards the leading edge plane of the interface formed by the low entropy incoming flow and the high entropy tip clearance flow of a rotor passage. This is referred to as the spillage of the tip leakage flow upstream of the rotor leading edge. The second critical feature was the backflow of fluid around the rotor trailing edge in the tip region. In the former flow feature, the trajectory of the interface within the rotor passage has been observed 92

107 experimentally to move towards the rotor leading edge plane as blade loading is increased (Saathoff and Stark 2000). Simulations done by Adamczyk, et al. (1993) showed that the interaction between the tip leakage vortex and the in-passage shock of a high-speed fan were linked to stall inception. It was concluded by the researchers that the vortex area increased as it crossed the shock, causing low-energy fluid to move forward towards the leading edge plane. This was the source of a large increase in endwall blockage that led to stall inception. Further information on the topic of rotor tip leakage spillage and spike-type stall inception is detailed by Tan, et al. (2009). The results of Koch s correlation indicated that the stall-limiting stages at 100% speed are likely stages 5 and 8, whereas the stall-limiting stages at part-speed N% are likely stages 1 or 2. The tip leakage flows of these stages were analyzed at near-stall simulations at both speeds to substantiate the results of Koch s correlation for stalling pressure rise as it was applied in this work. First, the tip leakage flows of rotor 5 are compared at design and N% speeds in Figure

108 Figure 6.12: Rotor 5 tip leakage flow, near-stall points at 100% speed (left) and part-speed N% (right) In Figure 6.12, the 100% speed near-stall simulation result is shown on the left, and the part-speed N% near-stall simulation result is shown on the right. A non-dimensional axial velocity is plotted, where regions of blue color indicate zero or negative axial velocities, corresponding to regions of reverse flow. The interface between the main flow and tip leakage flow is clearly seen near the leading edge plane. The trajectories of the interfaces in both simulations are quite close to the plane formed by the rotor leading edges. Careful observation of the 100% speed simulation shows its leakage flow trajectory is closer to spilling than the part-speed N% simulation. These results are compatible with the results of Koch s correlation, which estimated a stage 5 stalling pressure rise ratio of 99.4% for the 100% speed near-stall case and 94.8% for the N% speed near-stall case. Figure 6.13 shows a similar comparison for the tip leakage flow of rotor 8. 94

109 Figure 6.13: Rotor 8 tip leakage flow, near-stall points at 100% speed (left) and part-speed N% (right) Again, the 100% speed near-stall simulation is shown on the left and the part-speed N% near-stall simulation is shown on the right. Here we can see that the trajectory of the tip leakage flow is farther into the blade passage in the N% speed case as compared to the 100% speed case. This result is compatible with the results of Koch s correlation, which estimated a stage 8 stalling pressure rise ratio of nearly 99.9% for the 100% speed case, and about 80.0% for the N% speed case. For comparison, the tip leakage flows of stage 12 at near-stall points at 100% speed and at N% speed are compared. These correspond to a staling pressure rise ratio of about 71.6% and 24.2%, respectively. One would expect that the N% speed case should have its tip leakage vortex quite far from spilling, whereas the 100% speed case s tip leakage 95

110 vortex should be significantly closer towards the leading edge. These trends are reflected in Figure Figure 6.14: Rotor 12 tip leakage flow, near-stall points at 100% speed (left) and part-speed N% (right) These results indicate that Koch s correlation for stage stalling pressure rise seems to identify those stages in which the rotor tip leakage flow is nearly spilling ahead of the rotor leading edge. This spillage of the tip leakage flow has been related to stall onset by several researchers (Tan 2009). However, unlike the results of Vo (2001), there was no significant backflow observed around the trailing edge of the stall-limiting rotors in the current work. 96

111 7 Conclusions and Recommendations 7.1 Conclusions The stage stacking procedure described in the current work was shown to have the capability to estimate overall compressor performance well at off-design operating points for the 13-stage compressor which was the focus of the current work. At 97% corrected speed, the stage stacking estimate was within 1.6% of overall total pressure ratio and within 0.5 points in efficiency of the CFD result. This good agreement was due to the implementation of a blockage term which mathematically closed the equations used to estimate performance with the stage stacking procedure. This blockage term was shown to agree well with the blockage defined by Khalid (1999), which was based on displacement thickness due to endwall boundary layer growth. It was shown that the blockage parameter defined in the current work correlated well with stage inlet flow coefficient, similar to the stage performance parameters of ideal work coefficient and adiabatic efficiency. The stage performance curves used in the current stage stacking procedure were constructed using flow information extracted from APNASA CFD simulations of groups of stages. It was necessary to simulate stages in groups so that the near-stall and nearchoke portions of the performance curves could be populated, which is difficult, if not impossible, if all stages in the compressor are simulated at once. This is because of the way the stages are aerodynamically matched. When simulated as smaller groups, stages are able to perform at a larger range of flow coefficients as compared to the 97

112 range possible in simulations of the entire compressor, as demonstrated in the current work. This larger range of performance allows for the construction of the near-choke and near-stall portions of the performance curves, which are important to define when generating performance estimates far from design point. The current work demonstrated that the first two stages within these simulated groups of stages were adequate to behave as flow generators which established the multistage environment for subsequent stages. Furthermore, it was shown that there was a negligible shift in the performance curves of a stage upon the addition of the downstream stage within the simulation. The stage characteristics revealed a collapse to a single curve, irrespective of corrected speed, for stages 5 and beyond. This finding is in agreement with the literature, which indicates that the performance curves of a stage will collapse to a single curve when the stage inlet relative Mach number is less than 0.75, which holds true for stages 5 and beyond for the current compressor. For stages 1 through 4, which have inlet relative Mach numbers greater than 0.75, the stage characteristics were shown as distinct curves for different corrected speeds. Additional applications of the stage stacking procedure were demonstrated with operability and bleed sensitivity studies. These studies indicated that the minimum speed for operability with geometry and bleed rates fixed was a corrected speed referred to as N% speed. Attempts to estimate performance using the stage stacking procedure for part-speed N2%, which was a corrected speed 2% lower than N% speed, 98

113 indicated that the compressor will not operate at N2% speed with fixed bleeds and geometry. It was shown that the stage stacking procedure can be used to establish a new bleed schedule which would allow for stable operation at part-speed N2% by means of an increase to the stage 5 bleed rate. These stage stacking estimates on operability were corroborated by subsequent APNASA CFD results. These CFD results were generated by using the stage stacking performance estimates as an initial guess for CFD boundary conditions. Finally, the APNASA CFD simulations of the entire compressor were used to assess Koch s correlation for stage stalling pressure rise. This assessment indicated that those stages with rotor tip leakage flows close to spillage upstream of the rotor leading edge were the same stages which were near the maximum static pressure rise as estimated by the correlation. Thus, this correlation seems to be valid for stages which are thought to exhibit wall stall (as opposed to blade stall). The correlation indicated that front stages were stall-limiting stages at part-speeds, and that middle stages were stalllimiting stages at design speed. These results generally agree with the literature (Cumpsty 1989), and are reflective of the design intent of the compressor designer (Wasdell 2011). 7.2 Future Work At the outset of this work, a geometry was used which did not include blade fillets. Towards the end of this work, a mesh was made available which included fillets on all airfoils except rotor 1. This new mesh included fillets at both the hub and casing of the 99

114 IGV and VGVs, at the hub for rotors 2 through 13, and at the casing for cantilevered stators 4 through 13 and the OGV. The fillets were a feature of the actual compressor. The new mesh with fillets was generated such that it had an identical number of axial, radial, and tangential nodes as the original mesh without fillets. Stretching ratios in this new mesh were adjusted so that the axial index numbers of each airfoil leading and trailing edge were identical to the original mesh without fillets. This allowed for the restarting of existing simulations using the new mesh with fillets. The major effect of the inclusion of the fillet geometry was the unloading of rotor 1. This is illustrated in Figure 7.1, showing the circumferentially averaged total pressure profiles downstream of the rotor 1 trailing edge at design operating point for two meshes, one with fillets and one without fillets. The boundary conditions between these two cases were identical. 100

115 Figure 7.1: Normalized total pressure profiles aft rotor 1, with and without fillet geometry The unloading of rotor 1 had important implications on part-speed operation. This is because the front stages are generally the stall limiting stages at part-speed. A reduction in loading for these stages may increase the part-speed operating range of the compressor. Simulations at part-speed N% were generated using the new mesh with fillets. The exit boundary condition for this simulation with fillets was set to the nearstall value for the simulation without fillets. The flow features from these two cases at N% speed were compared in the following figures, which show contour plots of a normalized entropy function on axial planes downstream of the specified blade rows. The left-hand-side of the figures depicts the flow in the simulation without fillets, and the right-hand-side depicts the flow in a simulation using the new mesh with fillets. 101

116 Identical boundary conditions were applied for both simulations. Regions of red indicate high entropy flow, whereas regions on dark blue indicate low entropy flow. Figure 7.2: N% speed, entropy contours at rotor 1 trailing edge, no fillets (left) vs. fillets (right) Figure 7.3: : N% speed, entropy contours at stator 1 trailing edge, no fillets (left) vs. fillets (right) 102

117 Figure 7.4: : N% speed, entropy contours at rotor 2 trailing edge, no fillets (left) vs. fillets (right) The figures clearly show a significant reduction in high entropy flow in these front blade rows at part-speed N%. Recall that N% was the lowest part-speed for which simulations of the entire machine were generated without the addition of fillets. Attempts to operate at N2% speed, which was 2% corrected speed lower than N%, resulted in numerical stall while using nominal bleed rates, as was discussed in Section 5.1. It was found that the compressor would operate at part-speed N2% if the stage 5 bleed rate was increased 7 times greater than the nominal value for the mesh without fillets. Since this new mesh with fillets showed the unloading of rotor 1, as well as lower entropy flow in subsequent blade rows, an attempt was made to generate a part-speed CFD simulation at N2% using the mesh with fillets. The existing simulation at N2% with the increased bleed rate was used as an initial guess for this new simulation. The stage 5 bleed rate was decreased down to the nominal value, and the mesh with fillets was used. This simulation converged without numerical stall. This was an important finding, 103