NUMERICAL STUDY OF THE STABILITY OF EMBEDDED SUPERSONIC COMPRESSOR STAGES

Transcription

1 NUMERICAL STUDY OF THE STABILITY OF EMBEDDED SUPERSONIC COMPRESSOR STAGES By Severin G. Kempf THESIS SUBMITTED TO THE FACULTY OF VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING Dr. Wing Ng, Chair Dr. Clint L. Dancey, Committee Member Dr. Peter King, Committee Member JULY 21, 2003 BLACKSBURG, VIRGINIA c Copyright by Severin G. Kempf, 2003

2 NUMERICAL STUDY OF THE STABILITY OF EMBEDDED SUPERSONIC COMPRESSOR STAGES Severin G. Kempf Abstract A numerical case study of a multistage compressor with relative supersonic rotors is presented. The purpose of the investigation was to determine the flow instability mechanism of the UEET compressor and its relation to the rotor shock structure in the relative velocity reference frame. The computational study was conducted with the NASA code AD- PAC, utilizing the mixing-plane assumption for the boundary condition between adjacent, relatively-rotating blade rows. A steady, five-blade-row, numerical simulation using the Baldwin-Lomax turbulence model was performed, creating several constant speed lines. The results are presented, highlighting the role shock structure plays in the stability of the compressor. The shock structure in the downstream rotor isolates the upstream rotor from the exit conditions until the shock detaches from the leading edge. At this point the shock structure in the upstream rotor moves, changing the conditions for the downstream rotor. This continues with increasing pressure at the exit until the shock in the upstream rotor detaches from the leading edge. This event causes an instantaneous drop in the mass flow rate, initiating positive incident separation on the suction side of stator-two.

3 Acknowledgements I would like to gratefully thank my advisor, Dr. Ng, for giving me a wonderful opportunity to work on this project. He has helped boost my confidence in my work and given me room to make my own successes and failures. I would also like to thank Dr. Dancey and Dr. King for taking time and effort to serve on my advisory committee. This project was funded by NASA as a portion of contract NAS I would like to thank the contract monitor, Dr. Randy Chriss, for the funding and guidance of this project. I would also like to thank Stephen Guillot and Sarah Stitzel for their technical and writing advice. A special thanks to Dr. Steve Wellborn at Rolls-Royce, Indianapolis for his wealth of knowledge. He was always willing to donate time and advice. It was greatly appreciated. I would like to thank my family, especially my parents, for their unconditional love and understanding. I am, who I am, because of you. Finally, I would like to thank my fianceé, Amy Switalski. Without you and your undying support, your understanding for my long nights in front of the computer, and for those special days you brought dinner to me I would not have accomplished so much. You are my love. Praise the Lord, Jesus Christ. Severin Kempf Blacksburg, Virginia August, 2003 iii

4 Table of Contents Abstract Acknowledgements Table of Contents List of Tables List of Figures Notation ii iii iv vi vii x 1 Introduction 1 2 Literature Review Background Supersonic Inlet Stall Surge Transonic Compressors Shock Structure Forward Sweep Studies Transonic Rotor Stability Margin UEET Compressor Design Computational Methodology Flow Solver Grid Generation Geometry Mesh Setup Boundary Conditions and Turbulence Model iv

5 3.3.1 Inlet / Exit Boundary Conditions Viscous Walls Patch Mixing-Plane Turbulence Model Solution Starting and Convergence Solution Post-Processing Visualization Solution Averaging Computational Results NASA Baseline Benchmark Embedded Stage Characteristics Shock Structure Stability Limit Failure Mode Conclusions 77 A ADPAC Non-dimensionalization 79 B Adjacent Blade Relative Motion Boundary Condition 80 C Multistage Compressor Off Design Performance 82 Bibliography 85 Vita 88 v

6 List of Tables 2.1 UEET compressor design mass flow rate and speed UEET compressor design total pressure ratio and efficiency UEET compressor blading summary Summary of reference conditions for UEET simulations UEET two-stage compressor mesh density summary Average y+ values for UEET two-stage grid Mesh parameters for UEET stator-two mesh density study vi

7 List of Figures 1.1 Supersonic airfoil shock structure schematic Target goal of UEET compressor program Pictorial representation of the changing shock structure in a supersonic inlet Graphical representation of supersonic inlet unstart Supersonic airfoil shock-structure schematic with increasing back pressure Transonic stage compressor map Meridional view of the UEET compressor flow path showing the blade row locations UEET inlet guide vane geometry UEET rotor-one geometry UEET stator-one geometry UEET rotor-two geometry UEET stator-two geometry Three-dimensional finite volume cell Example airfoil Example airfoil passage Example bounding surfaces for individual passage Close up of typical leading edge mesh Meridional view of UEET two-stage compressor mesh Isometric view of UEET two stage compressor mesh Stator-two mesh study comparison of radial distribution of exit total pressure 34 vii

8 3.9 Stator-two mesh study comparison of radial distribution of exit axial velocity Stator-two mesh study comparison of radial distribution of exit tangential velocity Rotor-two mesh density comparison of UEETS1 mesh and fine mesh Mach number contours Two-dimensional representation of phantom cells Boundary condition summary Patch boundary condition illustration Mixing-plane data transfer Mixing-plane circumferential variation Turbulence model comparison Example mass flow rate convergence history Example non-dimensional mass flow rate convergence history Example error residual convergence history Example pressure ratio and efficiency convergence history Three-dimensional visualization Rotor inlet profile Stator inlet profile Predicted stage-two performance characteristic of UEET compressor at two rotational speeds Predicted stage-one performance characteristic of UEET compressor at two rotational speeds Predicted overall performance characteristic of UEET compressor at two rotational speeds Mach contour with exit condition P s,exit /P 0,inlet = Mach contour with exit condition P s,exit /P 0,inlet = Mach contour with exit condition P s,exit /P 0,inlet = Mach contour with exit condition P s,exit /P 0,inlet = Mach contour with exit condition P s,exit /P 0,inlet = viii

9 4.11 Mach contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour with exit condition P s,exit /P 0,inlet = Static pressure contour at 103% shaft speed Stage characteristic comparison using static to total pressure ratio Total pressure plot before rotor-one unstart Total pressure plot after rotor-one unstart Rotor-two constant axial direction plane near trailing edge Stator-two constant axial direction plane near trailing edge Stage stability margin Influence of unstart on mass flow rate for UEET two-stage compressor C.1 Subsonic multistage compressor characteristic ix

10 Notation Acronyms ADPAC CFD LE MRQS PS RANS SS TANH TE UEET Advanced Ducted Propfan Analysis Codes, developed by Dr. Ed Hall at Rolls-Royce Indianapolis under NASA contract Computational Fluid Dynamics Leading Edge Monotonic Rational Quadratic Spline, a grid point distribution function in Gridgen Pressure Side Reynolds Averaged Navier-Stokes equations Suction Side Hyperbolic Tangent, a grid point distribution function in Gridgen Trailing Edge Ultra Efficient Engine Technology Symbols ṁ c p corrected mass flow rate, ṁ c = ṁ T 0 /T 0,std P 0 /P 0,std static pressure P 0 total pressure, defined by an isentropic relation on a point to point basis, P 0 = p (T 0 /T ) γ/(γ 1) P 0,std SM standard day total pressure ( lbf ft 2 ) stability margin, SM = P R SL ṁ c,dp P R DP ṁ c,sl 1 x

11 SM m modified stability margin, SM m = P R/P R DP (ṁ c/ṁ c,dp )(N c/n c,dp ) 1 T static temperature T 0 total temperature, defined by the relation T 0 = T + V 2 2C p T 0,std U v i V standard day total temperature (518.67R) local blade velocity denotes i th component of velocity in the absolute frame velocity magnitude in the absolute frame Greek φ axial velocity ratio, v z /U or v z /U tip Ψ T s static pressure rise coefficient, P s P 0 / 1 2 ρu 2 η pc ρ Ω polytropic efficiency static density engine shaft speed Subscripts () z denotes axial direction in compressor () r denotes radial direction in compressor () θ denotes circumferential direction in compressor () rel relative reference frame () 0 denotes a stagnation or total quantity () DP denotes design point () SL denotes stability limit xi

12 Chapter 1 Introduction To meet market demands, the gas turbine engine industry, like all other industries, is constantly pushing the current design limits. Engine companies want to produce engines that are stronger, lighter, and cheaper to run and manufacture. To achieve the aerodynamic and thermodynamic demands, engines continue to be designed with higher thrust to weight ratios and greater efficiency. Each new generation of compressors is designed for higher loading at a greater efficiency than the previous generation. Higher loading implies less stages for a given pressure ratio, which results in producing lighter compressors, thereby helping to increase the engine thrust to weight ratio. Two ways to achieve higher pressure ratios in fewer stages are illustrated in Equation 1.1. W ork ṁ = h 0 = Ω [(rv θ ) 2 (rv θ ) 1 ] (1.1) The specific work of the compressor is proportional to both the amount of turning ( v θ ) and compressor shaft speed (Ω). As the shaft speed is increased, portions of the rotor along the span transition from a relative subsonic to a relative supersonic inlet flow field. In the transonic regime, the relative supersonic sections increase the static pressure through shock diffusion, also causing flow turning in the absolute reference frame. Continuing to increase the rotational speed leads to supersonic rotors, where the inlet flow field is supersonic along the entire span of the rotor. 1

13 2 Relative Inlet Flow Oblique Bow Shock M >1 rel M <1 abs c M >1 rel Normal Passage Shock w M rel>1 U M <1 rel M <1 rel (a) (b) Figure 1.1: Airfoil cascade section showing oblique bow shock and normal passage shock for relative supersonic inlet flow. (a) Double shock system, near choke. (b) Single shock system, near stall. (adapted from [1]) A unified convention for the definition of transonic or supersonic compressor stages does not exist in the open literature. The definitions used in this thesis were adapted from a review of transonic fan compressor flow by Kerrebrock [1]. Kerrebrock defines transonic compressors as compressors exhibiting relative supersonic inlet flow at the tip and relative subsonic inlet flow at the hub. The axial component of the flow velocity is subsonic. In the upper radii sections, relative flow is diffused to subsonic velocities and turned in the rotor by both the shock and the blade camber. Further static pressure rise is generated in the following stator-row having primarily subsonic diffusion. Supersonic compressor stages are defined in this thesis as stages with a relative supersonic inlet flow to the rotor from hub to tip with the exception of endwall boundary layer regions. The relative flow is diffused to subsonic velocities and turned in the rotor by the shock and blade camber over the entire span of the rotor. The axial component of velocity is subsonic through the rotor. The rotor relative exit velocity is also subsonic. Both rotors in this study are supersonic rotors.

14 3 The tip section of transonic rotors and airfoil sections of a supersonic rotor have similar design motivations and shock structures in the relative velocity reference frame. In the tip region of a transonic fan, a double shock system exists near the choking point, as shown in Figure 1.1a. A single shock system forms as the compressor nears stall, as shown in Figure 1.1b. Kerrebrock suggested that operating with a single shock system is more efficient, but is closer to the stability limit. In the interest of stability, operation in the double shock region is usually chosen [1]. UEET Program Background The NASA UEET [2] (Ultra Efficient Engine Technology) program was envisioned to create a revolutionary technology leap in turbomachinery design. This research is part of the Highly Loaded Turbomachinery Technology Task. The program objective is to make an aggressive jump in the compressor design space of a magnitude beyond current design practices. Figure 1.2 shows graphically the UEET Compressor design in terms of technology achievement goals compared to current engines. The Average Work Factor (dh avg /Utip) 2 represents the average energy imparted to the flow and the polytropic efficiency (η pc ) represents the efficiency over an infinitesimal step of the compression process. Current highly-loaded compressor stages, having a high Average Work Factor, also have poor efficiency. The goal of the UEET program is to extend into the portion of the design space characterized by high-loading while maintaining good polytropic efficiency. The specific goal is to achieve a 12 : 1 compression ratio in four stages with an overall polytropic efficiency of 92%. The first two stages of the UEET compressor are supersonic stages. In computer simulations, the design reaches pressure ratio and efficiency goals. Upon further investigation, it was determined that a more fundamental understanding of embedded, supersonic stages in the multistage environment was needed. The objective of this thesis is to determine the role shock structure plays in the stability margin for multistage compressors with multiple supersonic rotors.

15 4 Polytropic Efficiency Engine Data Current Tech. Level UEET Average Work Factor (dh avg /U tip ) Figure 1.2: Target goal of UEET compressor program compared with core compressor data (adapted from [3])

16 5 A numerical study of the UEET compressor design was conducted to understand the role shock structure plays in the stability of the compressor. Relevant research and background will be presented in the Literature Review, Chapter 2. The setup of the numerical simulation, solver, and post-processing methods will be presented in Chapter 3, Computational Methodology. A baseline comparison and numerical results are presented in Chapter 4. Conclusions and Recommendations are presented in Chapter 5.

17 Chapter 2 Literature Review To understand the role shock structure plays in compressor stability for multi-supersonic stage compressors, a review of the fundamental mechanisms involved is beneficial. The literature review is broken down into a short background section, a review of transonic rotor blading, and a brief description of the NASA UEET design. The background covers the theory behind the shock structures in relative supersonic airfoil sections. Open literature contains no research relating the shock structure and stability limit in multiple, supersonic stages. Therefore, relevant transonic rotor research and a more detailed description of documented shock structures will be presented. Finally, the design methodology of the UEET compressor is presented for completeness. 2.1 Background As presented in the introduction, a very specific shock structure exists in the rotor passage, in the relative velocity reference frame. In turbomachinery, the relative frame is the frame of reference moving with the rotor at some shaft speed (Ω). These structures are analogous to those exhibited in supersonic inlets with a converging-diverging contour. The material regarding supersonic inlets is presented with a focus on the aspects relevant to supersonic rotors. This thesis also deals with stability limit considerations. To this end a brief definition of compressor stall is provided. The definitions of stall and surge are sometimes blurred 6

18 7 together, especially in reference to compressor performance maps. Surge is briefly defined only to provide a clear understanding of the focus of this research Supersonic Inlet Supersonic inlets cover a large area of research. Only a short treatment relevant to compressor rotor flow in the relative velocity reference frame is presented here. The material is based on Hill & Peterson s section on supersonic inlets [4, Pg 226]. Starting and unstarting of supersonic inlets is also clearly defined here for use with similar phenomenon in the rotor. Supersonic inlets are converging-diverging nozzles that are designed to efficiently diffuse the supersonic inlet flow to subsonic velocities. As supersonic flow enters a convergingdiverging inlet at design conditions (see i.e. [5], [6]), it decelerates in the converging section and accelerates aft of the throat. Figure 2.1 shows a pictorial representation of a convergingdiverging nozzle, with supersonic inlet conditions, for a range of exit static pressure conditions. Figure 2.1a illustrates a shock forming between the throat and the exit of the inlet. The shock is strong enough to increase the static pressure in the flow to match the exit static pressure. The shock position is stabilized by the exit static pressure; therefore changing the exit static pressure modifies the shock position. Flow upstream of the shock does not feel any effects of the changing back pressure, because pressure information cannot be communicated upstream in a supersonic flow. Increasing the exit static pressure will eventually cause a weak shock to stand in the throat as shown in Figure 2.1b. This is the peak efficiency point, the weak normal shock having the lowest possible shock losses. The converging-diverging inlet is acting as a supersonic diffusor, supersonically diffusing the flow from the inlet to the throat. Just before the throat, the Mach number is greater than unity. The weak shock reduces the Mach number below unity and subsonic diffusion takes place aft of the throat. Any further increase in exit static pressure results in the unstarting of the inlet, Figure 2.1c. Unstart is characterized by detached bow shock standing off the lip of the inlet that is strong enough to cause the flow field to be subsonic entering the converging-diverging contour. A weak normal shock

19 8 A i At M>1 M>1 M<1 (a) M>1 Weak Shock M<1 Detached Bow Shock (b) M>1 M<1 M=1 M<1 (c) Figure 2.1: Pictorial representation of the changing shock structure in a supersonic inlet (adapted from [4]). (a) through (c) represent increasing exit static pressure

20 9 cannot be positioned at the throat from an unstarted condition by reducing the exit static pressure to the peak efficiency point. A hysteresis exists such that the static pressure must be reduced to a value less than at the peak efficiency point, such as Figure 2.1a. The shock in the diverging section can then be repositioned to the throat by increasing the exit static pressure, illustrated in Figure 2.1b. The most important consequence can be illustrated in a graphical depiction of Figure 2.1. A plot of the mass flow rate and exit static pressure for varying inlet Mach numbers is shown in Figure 2.2. The characteristic is plotted for varying inlet Mach numbers (M inlet = 1.6, 1.7, 1.8), while holding the inlet static pressure and temperature constant. Figure 2.2a-2.2c show points corresponding to Figure 2.1a-2.1c for a constant inlet Mach number. The dashed line between Figure 2.2b-2.2c is the instantaneous change in mass flow rate at unstart. With increasing inlet Mach number, the instantaneous drop in the mass flow rate also increases. Given a high enough inlet Mach number, the exit static pressure required to unstart the inlet can be high enough to cause reverse flow in the inlet when unstart occurs. The passage shock in the rotor reacts to changing back pressure the same way as the supersonic inlet. Although the shock structure is more complicated in a relative-supersonic compressor airfoil, changing the exit static pressure causes a change in the shock position and structure. Information cannot travel upstream of the shock, with the exception of the subsonic boundary layer regions. This research will show that supersonic rotors exhibit a similar hysteresis and unstarting phenomenon analogous to the supersonic inlet Stall The stability margin of a compressor is usually limited by some form of stall. On an airfoil, stall is simply a region of separation, or region of low momentum fluid. In a rotor or stator passage, this region is felt by the upstream flow as a blockage, or an increase in the downstream static pressure. A variety of flow conditions can cause separation, including incident angle deviation, highly adverse pressure gradients, and shock induced separation.

21 A 1 /A t =A e /A t =1.25 p s /p ref =1.0 t s /t ref =1.0 M 1 =1.6 M 1 =1.7 M 1 = (a) (b) m/ m ref (c) P s,exit /P ref Figure 2.2: Graphical representation of supersonic inlet unstart based on derivation by Guillot [7] and author. (a) through (c) represent increasing exit static pressure In a rotor or stator passage, separation of one blade changes the inlet conditions to surrounding blades. This starts a process of rotating stall, where one stalled passage or set of passages rotates opposite the direction of shaft rotation at a fraction of the shaft speed. Experimental research by Day on a low speed compressor with four identical stages suggests stall starts with a small disturbance, usually near the tips of the first stage rotor, and spreads both axially and radially [8]. Success in increasing the stability margin was achieved by increasing the axial momentum near the tip of the rotor [8], [9]. Blowing on the rotor has been used with success on many low speed and subsonic compressors. The operating mechanism is to delay the onset of the emerging stall cell at the tip by decreasing the incidence angle near the tip at low values of the flow coefficient. Success has also been shown by blowing on transonic fans in a similar manner [10], but the operating mechanism is different and is related to the shock structure and overall momentum deficit. Research pertaining to shock structure and stall will be presented in Section 2.2.

22 Surge Many treatments of stall and surge use these terms interchangeably. Compressor maps usually show the operability limit line as the Stall or Surge Line. In this thesis, the operability limit is referred to as the Stability Limit. Surge is not the focus of this thesis, but a brief definition is presented to clear any confusion of the terminology. Surge is a cyclical change in the direction of the air flow from normal flow direction to reversed flow. It is a violent condition for the blades in the compressor, causing excessive force and vibration, which can lead to failure. Surge is a function of the design of the compressor and the downstream components of the engine. Unlike stall, surge is a system phenomenon. It is believed that surge always starts with stall inception, usually near the first stage rotor tips [8]. Most techniques for extending compressor operational range limited by surge focus on preventing or suppressing rotating blade stall. 2.2 Transonic Compressors Transonic fan and compressors have been a topic of research for many years. The shock structures in transonic rotors are similar to that in supersonic rotors and are therefore relevant to this study. A clear description of the shock structure is presented first. Recent literature has presented increased stability margin related to forward-swept transonic rotors. A look into the mechanism involved is reviewed. Finally, some open literature information about transonic fan performance and stability is presented Shock Structure Many publications have documented the shock structure resulting in fan and compressor rotors with transonic inlet conditions. Few have made concrete links between this shock structure and stall. Books such as Hill and Peterson [4] and Laksminarayana [11] outline the basic shock structures exhibited in compressor rotors with relative supersonic inlet flow much like Kerrebrock s [1] review mentioned in the introduction.

23 12 Strazisar defined the shock structure in more detail with an experimental investigation of a transonic fan rotor [12]. Laser anemometry was used for data collection. The passage shock existed at and below peak efficiency pressure ratios and was normal to the flow through the passage. Starting at choked pressure ratios, increasing the exit static pressure caused the oblique bow-shock angle to decrease, or become more normal to the flow and therefore stronger. The standoff distance of the oblique shock also increased with increasing back pressure from the peak efficiency point to the stability limit. Spanwise lean of the shock in the meridional plane was seen to cause radial flows. Broichhasen also documented similar structures with experimental and numerical streamline curvature calculations of a supersonic rotor cascade [13]. At increased back pressure conditions, a standing normal shock much like in a two-dimensional nozzle existed in the passage, stabilized by the geometry and exit static pressure. Both Schreiber [14] and Küsters [15] documented experimental and numerical results, respectively, of single transonic rotor cascades. They noted, as does Laksminarayana [11], that for this flow condition, the upstream leg of the oblique bow shock sets the inlet flow angle. This is called the unique incidence condition, whereby the inlet flow angle is not a function of downstream conditions but is set by the upstream shock. Expansion waves then turn the flow parallel to the surface. This is valid only for flows that are still supersonic after the oblique shock. Using the above literature review of inlets and transonic rotor shock structure along with the numerical simulations of the UEET two stage compressor, Figure 1.1 was modified as shown in Figure 2.3. In the new figure, the changing shock structure is related to increasing exit static pressure for a relative supersonic airfoil section. In Figure 2.3a, the incoming supersonic flow is decelerated through the weak oblique shock at low exit static pressures. The flow supersonically diffuses to the throat. Since the flow is still supersonic, it accelerates in the diverging portion of the passage. A strong passage shock reduces the flow to subsonic velocities. As the exit static pressure is increased, the passage shock moves upstream toward the apparent throat, becoming weaker. Peak efficiency occurs near the merging of the passage shock and the passage leg of the oblique shock as in Figure 2.3b. Further increase in exit

24 13 Inlet Flow M abs<1 c Weak Oblique Bow Shock Strong Oblique Bow Shock w M >1 rel U M >1 rel Passage Shock M <1 rel M <1 rel Increasing Exit Static Pressure (a) (b) (c) Figure 2.3: Schematic of changing shock structure of supersonic airfoil section with increasing back pressure (a) low exit static pressure, (b) near peak efficiency, (c) near stall (adapted from UEET research and literature, i.e. [1],[11],[16]) static pressure results in rotor unstart, shown in Figure 2.3c. Unstart is characterized by a strong oblique shock that is detached from the leading edge. The flow field entering the passage is subsonic and accelerates through the converging section and decelerates through the diverging section. In a numerical study of compressor casing treatments for high-speed compressor rotors in a real engine, Hall [17] noticed a link between increased stability margin and shock structure. Casing treatments that extended the compressor stability margin constrained the passage leg of the oblique shock deeper in the passage. Other ways of changing the geometry to increase the stability margin have also been tried, such as sweeping the rotor spanwise profile Forward Sweep Studies Similar to the idea behind casing treatments that extend the stability margin, researchers have found forward sweep can manipulate the tip-region shock structure in the same manner.

25 14 In Wadia s [18] single-stage experimental and numerical investigation of sweep, he observed that the radial loading distribution near stall was more uniform. At part speed, sweep had no aerodynamic advantage, implying that it was related to constraining the shock structure further back in the passage near the tip. Hah [19] backed this theory up with a numerical study of sweep of transonic rotors. He further noted that this radially uniform loading distribution prevented premature stall in the tip region. Due to inviscid effects, the shock must be perpendicular to the case in the meridional plane. Forward swept blading causes the shock to be constrained further back in the passage and, therefore, remain started. It was suggested that control of the shock structure in this manner reduced radial migration of low momentum fluid to the tip region, helping the tip performance. It was also noted by Hah [19] that when the passage shock moved upstream of the leading edge, the compressor became unstable. Wernet [20] noticed in a low speed compressor experiment that when the influence region of the tip vortex moved ahead of the rotor leading edge, the compressor became unstable. Denton [16] showed this to be true in transonic forward swept rotors as well. This has lead some researchers to believe that unstarting the tip of the rotor leads to the stability limit. Researchers have noted that the shock placement is important, but they have not concretely tied the shock position to some aspect of instability or the importance of it in the multistage environment Transonic Rotor Stability Margin Traditionally, the stability margin of compressors has been measured by some metric of flow range from the design point to the stability limit. Range in flow is typically the change in the corrected mass flow rate moving along a characteristic. A typical transonic compressor performance characteristic is shown in Figure 2.4. At partial corrected speed, the characteristic is the same as a subsonic stage characteristic. As the corrected speed is increased, there is an apparent reduction in the flow range, or stability margin.

26 Figure 2.4: Typical transonic stage performance map [4, Pg 297] 15

27 16 Wood and Strazisar conducted an experiment with a transonic rotor to determine the shock structure with laser anemometry [21]. They noted that near peak efficiency small perturbations in the mass flow rate resulted in large shock position movement. This sensitivity can be seen in the vertical characteristic at the higher speeds of the compressor map in Figure 2.4. Due to choking, large variations in pressure ratio can be seen with little to no change in the corrected mass flow rate. This has lead some researchers [19] to define the stability margin with respect to a change in both corrected mass flow and pressure ratio as in Equation 2.1. SM = P R SL ṁ c,dp 1 (2.1) P R DP ṁ c,sl Here P R and m c are the pressure ratio and corrected mass flow rate respectively, subscript SL is the stability limit point, and subscript DP is the design point. It is clear that a compressor design with relative supersonic inlet flow over the span of the rotor is unique with respect to the literature. The shock behavior will exhibit key features that are similar in nature to nozzle flow and the shock structure of transonic compressors. Although the shock structure of such machines has not been overlooked entirely, its fundamental role in stall inception and, in particular, multi-supersonic stage compressor performance has not been quantified in the literature numerically or experimentally. This research is an attempt to fill this void in the literature. 2.3 UEET Compressor Design The UEET compressor design process and design refinement is documented in the NASA UEET compressor design report [22]. The current two-stage model of the UEET compressor was designed in two distinct steps. The first step used traditional through flow design techniques to design a four-stage compressor. As mentioned in the introduction, the design goal was high efficiency and an elevated loading level. The design performance models were adjusted based on postulated technology advancements. ADPAC, a three-dimensional viscous code, was used on isolated blade rows to iteratively guide the design. Table 2.1 shows

28 17 the design parameters resulting from step one. Table 2.2 shows the design pressure ratio and efficiency for the first two stages of the four-stage design. Table 2.1: UEET compressor design mass flow rate and speed Mass Flow Rate, ṁ c,dp 70.0 lbm/s Case Diameter in Shaft Speed, Ω DP RP M Tip Speed, U t,dp 1477 ft/s Table 2.2: UEET compressor design total pressure ratio and efficiency Stage P R 0 η pc Overall NASA desired to test the design techniques that were being developed to design the UEET compressor. The first two stages, including the inlet guide vane (IGV), were chosen to go through a refined design process and be built for experimental tests. A three-dimensional inverse method was used to adjust the airfoil shapes from the original design, tailoring the loading distribution. As before, the performance indicator was high efficiency with high loading. The experimental UEET compressor rig is a proof-of-concept research compressor to test proposed technologies for off-design performance and increased stability margin. These technologies can then be added more accurately to the design models for the refined four stage design. It is important to understand the history of the refined two-stage design from the fourstage. Running the first two stages alone misrepresents the quality of the design. Had stage two been designed as the last stage of the compressor, it would be more robust against the changing exit conditions due to engine throttling. 1 Calculated from information in reference [22]

29 18 Table 2.3 summarizes some aspects of the final, refined, two-stage design. A dimensioned, meridional view of the flow path is shown in Figure 2.5. Hub, mean, and tip cross sections along with a three dimensional view of each blade is provided in Figures 2.6 through Table 2.3: UEET compressor blading summary Row Number of Blades Aspect Ratio Clearance IGV R S R S The behavior of relative supersonic blading is highly dependent on the design of the airfoil shape. The geometry controls shock position and the throat location, dictating the shock structure at peak efficiency. The UEET compressor rotor suction surfaces (Figures 2.7 and 2.9) are designed with a straight inlet section. The minimum area occurs in the passage near the leading edge. The aft section is designed for controlled subsonic diffusion. At peak efficiency, the oblique shock and passage shock are almost merged together because the minimum area is so close to the leading edge, similar to Figure 2.3b.

30 IGV R1 S1 R2 S Axial Distance [in] Radius [in] Figure 2.5: Meridional view of the UEET compressor flow path showing the blade row locations

31 20 LE TE Airfoil Arc Length [in] Hub Mean Case Axial Distance [in] Figure 2.6: UEET inlet guide vane geometry LE SS TE Airfoil Arc Length [in] Hub Mean Case Axial Distance [in] Figure 2.7: UEET rotor-one geometry

32 21 LE PS TE Airfoil Arc Length [in] Hub Mean Case Axial Distance [in] Figure 2.8: UEET stator-one geometry LE SS TE Airfoil Arc Length [in] Hub Mean Case Axial Distance [in] Figure 2.9: UEET rotor-two geometry

33 22 LE PS TE Airfoil Arc Length [in] Hub Mean Case Axial Distance [in] Figure 2.10: UEET stator-two geometry

34 Chapter 3 Computational Methodology This section presents the computational domain setup for the numerical simulations used to model the UEET two-stage compressor. It is assumed that the reader is familiar with the field of computational fluid dynamics applied towards turbomachinery, including the Reynolds Averaged Navier-Stokes equations, discretization of the equations, and the transformation of geometry to a computational grid. For the UEET two-stage simulation, the NASA code ADPAC was used. ADPAC stands for Advanced Ducted Propfan Analysis Codes and was developed at Allison Engine Company, Indianapolis under contracts from NASA Glenn Research Center. General information about ADPAC will be briefly described, focusing on information important to a user. The method used for grid generation for the discretized equations in ADPAC will be presented along with details of the mesh. Boundary condition specifications and processes related to running the solution with ADPAC will be discussed. Finally, methods used in post processing will be presented. 3.1 Flow Solver ADPAC is a general turbomachinery code for solving a three-dimensional, time dependant form of the Reynolds averaged Navier-Stokes equations. ADPAC uses a structured grid with multiple blocks, which allows for the modelling of complex geometry. The finite volume 23

35 24 formulation is integrated using a Runge-Kutta time marching scheme with added numerical dissipation. ADPAC is second order accurate in space using an explicit central difference finite volume scheme. It is explicit in the pseudo time step for steady calculations and implicit in real time for unsteady calculations. The algorithms in ADPAC were formulated in a non-dimensional manner, described in Appendix A. The user specifies the dimensional values in the ADPAC input file. The values of the reference temperature and pressure are set so that the non-dimensional value over the domain is approximately unity. This aids in the stability and convergence of the solution [23]. Table 3.1 summarizes the reference values for air, used for the UEET simulations. Table 3.1: Summary of reference conditions for UEET two-stage compressor numerical simulations Quantity Value Units R ref ft lbf/slug R P 0,ref lbf/ft 2 T 0,ref R To solve the turbulence closure problem, ADPAC employs the Boussinesq approximation [23], which treats the turbulent stresses as viscosity. This results in an effective viscosity term of the form of Equation 3.1. µ effective = µ laminar + µ turbulent (3.1) The particular turbulence model used in this simulation to set the effective viscosity will be discussed in Section

36 25 i,j+1,k+1 i+1,j+1,k+1 i,j+1,k i+1,j+1,k i,j,k+1 k i,j,k j i Q i,j,k i+1,j,k+1 i+1,j,k Figure 3.1: Three-dimensional finite volume cell (adapted from [23]) The flux and stress terms are calculated at the finite volume cell face and the dependent variables are stored in the cell center. This is illustrated in Figure 3.1. The vector of dependent variables is shown in Equation 3.2 ρ ρv z Q = ρv r ρv θ ρe t (3.2) where v z is the axial component, v r is the radial component, and v θ is the circumferential component of velocity. The total energy term is defined as e t = p (γ 1) ρ + 1 ( v 2 2 z + vr 2 + vθ) 2 (3.3) This numerical case study utilized the steady calculation mode within ADPAC. The time step is calculated locally, using the largest value of the time step possible for each cell. This destroys any meaning to the transient behavior leading up to solution convergence. However, the unstarting of the rotor is felt to be valid. In general, larger cells occur in the inviscid regions such as the flow in the center of the passage. These cells will have approximately the

37 26 same time step. Smaller cells occur mostly in the boundary layer or viscous regions. The smaller cells have a smaller time step and will converge slower than the larger, inviscid cells. Because this compressor has little to no shock induced boundary layer separation over most of the operating line, the changing shock structure can be considered an inviscid phenomena. For this reason, the trends are still representative of the actual changing shock structure in the compressor. ADPAC has been extensively tested and is used by many research labs and companies both for design and analysis. Specific information on the governing equations, derivation of the discretization, and smoothing schemes can be found in the ADPAC User s Manual [23] and NASA contract reports related to the development and verification of adpac such as [17]. 3.2 Grid Generation The discretization in ADPAC is formulated for a structured computational domain. Grid generation is a transformation of the real geometry to a rectangular 1 geometry of discrete points. The geometry provided by NASA was organized in a form specifically convenient for turbomachinery applications. The geometry was manipulated into bounding surfaces of the blade row passage volume with a program written by the author. The interior grid points were then generated using Gridgen Geometry The geometry for the UEET two-stage compressor was provided by NASA. The hub and case profiles are defined as points in the axial (e z ) and radial (e r ) directions. Each blade row is defined by a stack of airfoil sections, defined by z, r, θ 1, and θ 2 point triplets in cylindrical coordinates. In this format, θ 1 and θ 2 define the pressure and suction surfaces of the airfoil along a common axial position and radius. 1 rectangular prism in three-dimensional space 2 Gridgen c is a general commercial grid generation package created by Pointwise

38 27 Airfoil Arc Length [% Axial Chord] Pressure Surface Suction Surface Axial Chord [%] Figure 3.2: Example of airfoil pressure and suction surface geometry specification, UEET stator-one mean airfoil Figure 3.2 shows an example of an airfoil section. The pressure and suction surface splines are monotonic in the axial direction (e z ). This allows a regular mesh gridding in the axial direction. The geometrical representation in this form allowed an easy transformation to a blade passage. One surface, the pressure surface for example, was displaced by one pitch (pitch = numberofblades/2π) in the circumferential direction (e θ ). This is illustrated in Figure 3.3. Each airfoil section in the stack for each blade row was manipulated in this manner to form the solid surface boundaries of the blades. After setting up the blade surface boundaries for each blade row, a boundary was created dividing the blade rows into individual blocks. The front and rear boundaries for these blocks was found by an axisymmetric spline, interpolated in the spanwise direction, dividing the blade rows. The spline was rotated circumferentially to form the computational boundary plane between two consecutive blade rows. The inlet section of the IGV was defined as

39 28 Airfoil Arc Length [% Axial Chord] Pressure Surface Suction Surface Axial Chord [%] Figure 3.3: Example of airfoil passage created from pressure and suction surface geometry, UEET stator-one mean airfoil a constant axial plane, 1.5 axial hub chords forward of the IGV. In a similar fashion the stator-two exit plane is 1.5 axial hub chords downstream of the stator-two trailing edge. Both rotors have a clearance gap between the rotor tip and the engine case. The case curve was displaced in the radial direction (e r ) by the clearance specification. Blade surface splines were intersected with the displaced case spline and split, forming separate bounding surfaces for the blade surface and the area in the clearance gap. The hub curve was displaced in the same manner to form the hub clearance gap for the cantilevered stators. Assembled together, these surfaces form a fully closed bounding box for each blade row passage. They were imported into Gridgen and used as fixed surfaces to attach an internal grid of points. Figure 3.4 shows an example assembly of the bounding surfaces. ADPAC handles multiple blocks. A mesh was created for each blade row using the bounding surfaces described above. Each mesh was combined to form a multiple block mesh. There are no overlapping mesh boundaries. Each boundary must have a boundary condition specification that will be discussed later in this chapter.

40 29 Gap Case Inlet Blade Exit Hub Figure 3.4: Example bounding surfaces for individual passage, UEET rotor-one Mesh Setup The mesh size and spacing was dictated largely under the guidance of an industry expert, Dr. Wellborn of Rolls-Royce, Indianapolis [24]. The bounding boxes created as described above where modelled after an example grid provided by Wellborn for the same geometry. A standard sheared H-mesh was used for each blade row in the UEET two stage compressor model. Each blade surface is defined with 65 points in the axial direction and 65 points in the radial direction. The inner blade rows have 25 points from inlet to leading edge and 25 from trailing edge to exit plane. The IGV inlet section and stator-two exit section both have 61 extra points. The inner blocks have approximately 360, 000 mesh cells. The IGV and stator-two have approximately 475, 000 cells, because of the extra inlet and exit regions respectively. This gives an overall mesh size of just over two million cells. Table 3.2 is a summary of the computational size used for each individual blade row.

41 30 Table 3.2: UEET two-stage compressor mesh density summary, where i is axial, j is radial, and k is circumferential directions Row Mesh Size Clearance i j k Gap IGV R S R S The clearance gap for each blade row, given in Table 2.3, was discretized in the radial direction with 5 mesh points. Equal spacing was used inside the gap, setting the case, near wall spacing at for rotor-one and for rotor-two. For mesh conformity in the spanwise direction, the same near wall spacing was used at the hub. The near wall spacing for both stators was The Monotonic Rational Quadratic Spline (MRQS) 3 [25] distribution was used for the points in the spanwise direction between the near wall spacings from hub to case. The blade leading edge and trailing edge spacing was set to approximately 0.1% axial chord, measured at the hub. A hyperbolic tangent distribution was used to distribute the points from the inlet to the leading edge, the leading edge to trailing edge, and trailing edge to the exit. The distribution was along the axial direction rather than blade surface arclength. Spacing on the blade surface in the circumferential direction was set at 0.03% axial chord at the hub. Table 3.3 shows the average y+ values for each blade row. Again the hyperbolic tangent distribution was used to distribute the points in the circumferential direction. A close up of the leading edge of the rotor-one mesh at 50% span is shown in Figure 3.5. The figure illustrates the leading edge and blade surface spacing. 3 The MRQS distribution has better interior spacing than the TANH distribution, but it has undesirable endpoint spacing. Since the endpoint spacing was set by the clearance gaps, MRQS was a better choice.

42 31 Table 3.3: Average y+ values for UEET two-stage grid Blade Avg y+ Values Row Blade Surf. Endwalls IGV R S R S Figure 3.5: Close up of leading edge rotor-one mesh, near hub of UEET two-stage compressor

43 32 IGV R1 S1 R2 S2 Figure 3.6: Meridional view of UEET two-stage compressor mesh Figure 3.6 shows a meridional view of the complete two-stage UEET compressor with IGV. In this figure, the concentration of points at both the hub and case near walls can be seen as well as point concentrations at the leading and trailing edge of each blade row. Figure 3.7 shows an isometric view of the mesh. One blade per row is shown with the pressure side tinted blue and the suction side tinted red. Multiple passages are shown along the hub, illustrating the root blade shape and spacing in both the axial and circumferential directions. Several grids prior to the final mesh were constructed with varying methods and sizes. The grid quality and solution independence was then reviewed by Wellborn [24]. Following this initial review, the grid construction methodology presented above was developed. Because of the complexity of the problem and available computer resources, a formal mesh independence study was not conducted. The author is unaware of such a study for a computational problem of this magnitude. However, the overall size and spacing of the mesh, coupled with the choice of turbulence model are very similar to those run in industry on a daily basis [24] and similar academic studies [26], [27], [19], [16]. An independent review of the final mesh by Wellborn suggested flow solution independence would occur through out most of the domain, with exceptions very near both endwalls. To back this up, a limited study of the stator-two mesh was performed. Table 3.4 summarizes the four grids. The UEETS1 grid was used for this thesis.

44 33 R2 S2 PS R1 S1 PS SS IGV SS Figure 3.7: Isometric view of UEET two stage compressor mesh Table 3.4: Mesh parameters for UEET stator-two mesh density study Mesh Axial Radial Circ. Axial Total % change Points Points Points (on Blade) from UEETS1 Coarse L ,869 57% Coarse L ,821 35% UEETS ,565 FINE L , % To compare the four grids, exit profiles were axisymmetrically mass averaged at the same axial location. Figure 3.8 shows a comparison between the exit total pressure distribution for each grid. Figures 3.9 and 3.10 show the axial and tangential velocity distributions, respectively. Each figure shows tat the two coarser grids had poor resolution along the spanwise direction, especially in the endwall regions. Both the UEETS1 and fine grid show good agreement. To address shock resolution issues, a fine rotor-two grid was constructed and compared to the UEETS1 rotor-two grid. The fine grid had a 25% increase in density along the blade surface in the axial direction. Figure 3.11 shows the comparison. The upper passage of

45 Coarse L2 Coarse L1 UEETS1 Fine L1 Span [%] P 0,exit /P ref Figure 3.8: Stator-two mesh study comparison of radial distribution of exit total pressure Coarse L2 Coarse L1 UEETS1 Fine L1 Span [%] v z [ft/s] Figure 3.9: Stator-two mesh study comparison of radial distribution of exit axial velocity

46 Coarse L2 Coarse L1 UEETS1 Fine L1 Span [%] v θ [ft/s] Figure 3.10: Stator-two mesh study comparison of radial distribution of exit tangential velocity Figure 3.11 is the UEETS1 grid and the lower passage is the finer grid. No discernable difference can be seen between the two solutions. Increasing the points in the passage results in better resolution of the shock, but the shock position, acceleration regions, and diffusion regions remain the same. 3.3 Boundary Conditions and Turbulence Model This section presents the boundary conditions used for the ADPAC model of the UEET two-stage compressor. Each boundary condition will be briefly explained. The choice of turbulence model will also be presented. Each boundary condition is imposed to set specific values at a physical or mesh boundary. In a cell centered formulation, this is accomplished with the use of phantom cells. Phantom cells are fictitious cells padding the mesh outer boundaries. Figure 3.12 shows a

47 36 UEETS1 R2 Fine Mesh Figure 3.11: Rotor-two mesh density comparison of UEETS1 mesh and fine mesh Mach number contours

48 37 j Boundary Conditions on Mesh Boundary Mesh Vertices Cell Centered Data i Phantom Cells Figure 3.12: Two-dimensional representation of phantom cells (adapted from [23]) two-dimensional mesh with cell centered data and phantom cells. The boundary conditions are applied by using the phantom cell values to manipulate the fluxes at the boundary. For reference, a summary of a majority of the boundary conditions is illustrated in Figure The following sections explain each of the boundary conditions in greater detail Inlet / Exit Boundary Conditions For the inlet boundary condition, ADPAC s Turbomachinery Inflow Boundary Condition (INLETT) was used. The INLETT boundary condition imposes a user specified radial distribution of total pressure, total temperature, radial flow angle (β r ), and circumferential flow angle (β θ ) at the inlet. The method of characteristics is used to calculate the phantom cell velocities. Phantom cell values of the five dependent variables are then calculated. Only the user specified radial distributions are circumferentially constant in the phantom cells. Surrounding cell centered data influences the dependant variables both in the phantom and first interior cells, causing circumferential variation.

49 38 (e) Periodic (e) Blade Surface (c) Inlet (a) (e) Hub (d) (e) Exit (b) a) Po, To, Angle Distribution b) Static Pressure c) Viscous Surface d) Moving Viscous Surface e) Periodic Figure 3.13: Boundary condition summary for individual example blade row A constant spanwise inlet profile was used at the inlet with the following values P 0 /P 0,ref = 1.0 T 0 /T 0,ref = 1.0 β r = 0.0 β θ = 0.0 The exit boundary condition, Turbomachinery Exit Boundary Condition (EXITT), is formulated in a similar manner. At the exit, only the exit static pressure is needed, specified at either the hub or case. Radial equilibrium is then utilized to integrate the pressure radially. The density is based on an isentropic relation of the near boundary and phantom cell pressures. The method of characteristics is again used to extrapolate the phantom cell velocities. The five dependent variables are then set for the phantom cells. Again, it should be emphasized that setting an exit static pressure does not mean the exit static pressure is

50 39 constant over the physical or computational exit plane. Because of the radial equilibrium integration, the static pressure in the phantom cells varies radially [23]. The numerical study presented in this thesis is simulating the throttling of the engine. This is accomplished by computing solutions at a range of exit static pressure that were integrated radially from hub to case. For the 101% speed simulation, solutions were calculated for a range from P s,exit /P 0,ref = 3.25 to The design pressure ratio occurred at a ratio of Viscous Walls Solid surfaces, including the hub, case, and blade surface boundaries, use ADPAC s Solid Surface Viscous No-Slip Boundary Condition (SSVI). SSVI is a viscous wall boundary condition allowing a wall rotational speed and wall temperature. A solid surface is defined as having no convective flux through the surface. This is represented by V ˆn = 0 (3.4) where ˆn is the surface outward normal vector and V is the velocity vector. The phantom cell velocities are calculated so that the convective flux is zero through the surface. Viscous surfaces have the added specification of the no slip condition. Because the surface can have its own rotational speed, the relative velocity must be zero. V rel = 0 (3.5) Here V rel is the relative velocity vector, V rel = [v z, v r, v theta rω] (3.6) and Ω is the rotational speed of the surface. The pressure of the phantom cell is set by the simplified momentum equation in the normal direction, stating that the pressure gradient normal to the surface must be zero. p n = 0 (3.7)

51 40 Patch Phantom Cells Mesh 1 Mesh 2 Mesh 1 Mesh 2 Interior Cells Figure 3.14: Patch boundary condition takes interior cell data and patches it into phantom cell of adjacent block For this simulation, the viscous walls were run as adiabatic walls. Both the density and energy are then set assuming no heat flux through the surface. Therefore the temperature gradient normal to the surface must be zero Patch T n = 0 (3.8) Modelling only one passage for each blade row assumes that each passage is periodic around the annulus. To simulate this the constant circumferential sections leading up to the blade surface from the inlet are then patched together as illustrated by arrow e in Figure A patch boundary condition patches the last interior cell of one mesh to the phantom cell of the adjacent mesh and vice versa, as shown in Figure Likewise the circumferential sections from the blade trailing edge to the exit are patched together. The clearance gap for the rotors and stators is also modelled with a patch. The last four cells (five grid nodes) in the radial direction of the rotor-one pressure surface are patched to the last four cells of the rotor-one suction surface. The same is done for rotor-two and the first four cells of stator-one and stator-two. This is sometimes refereed to as the hyperspace clearance gap model. It assumes no blade thickness and therefore no boundary layer development in the clearance gap. This boundary condition is used extensively in the design systems in industry and provides relatively good agreement between model and experiment [28].

52 Mixing-Plane To model the boundary between the adjacent blade rows, the Mixing-Plane model was used. The mixing-plane model passes circumferentially-averaged spanwise data from one blade row to the next. It allows the transfer of average blade wake data to the adjacent row or the average presence of a downstream row to the upstream row. In ADPAC, the mixing-plane boundary condition Multiple Block Circumferential Averaging Routine for Multiple Blade Row Turbomachines (MBCAVG) circumferentially area averages information from the neighboring block for input into the current block. boundary condition area averages the near boundary plane of cell centered data of dependent variables from the adjacent block in the circumferential direction. The corresponding plane of phantom cell values in the current block are simply replaced by this circumferentially averaged span wise distribution of the five state variables 4. This is illustrated in Figure Only the phantom cells hold circumferentially constant data. The first interior cell from the boundary shows circumferential variation from the presence of surrounding information. This is illustrated in Figure The overall model in ADPAC utilizing the mixing-plane models an inherently unsteady problem with a steady assumption. The The unsteady behavior of relative motion between blade rows and unsteady wake sheading are not captured (see Appendix B). Instead, only spatially-averaged data is transferred between adjacent blade rows. In the physical domain, other effects will influence the solution, which could only be resolved with an unsteady analysis. The analysis presented here focuses on the trends of shock structure movement and the resulting effect on the rest of the system. Unsteady losses will cause the shock position to also become unsteady. It is assumed in this thesis that a steady calculation will be representative of the mean shock position, allowing for a meaningful study of the trends resulting from engine throttling. 4 In ADPAC, static pressure is carried along in the calculation as a 6th state variable. In routines with data averaging, the static pressure is averaged instead of energy and the energy term is backed out per Equation 3.3

53 42 Phantom Cells Rotor Rotation Rotor Mesh Stator Mesh Mixing Plane Figure 3.15: Mixing-plane passes circumferentially-averaged data from neighboring row to phantom cells (adapted from [26]) r Rotor 1 Exit Plane Stator 1 Inlet Plane Figure 3.16: Rotor-one exit plane and stator-one inlet plane showing circumferential variation in first interior cell after mixing-plane boundary condition

54 43 Figure 3.17: Comparison of efficiency prediction between various turbulence models and experimental data in a transonic fan [29] Turbulence Model As mentioned in Section 3.1, ADPAC utilizes the Boussinesq approximation [23]. All turbulence models in ADPAC, therefore, manipulate the value of effective viscosity for each cell centered point. ADPAC has a choice between a mixing length, the Baldwin-Lomax (zeroequation), Spalart-Allmaras one-equation, and the k R two-equation turbulence models. Turner and Jennions compared various turbulence models for a transonic fan numerical simulation [29]. Figure 3.17 shows a spanwise distribution of predicted adiabatic efficiency. The figure compares the two-equation turbulence model and several implementations of the Baldwin-Lomax model with experimental data. Both the standard and modified coefficient Baldwin-Lomax model incorporating wall functions compares well with the experimental data. Wall functions are used to model the near wall flow without discretizing the inner boundary layer. Because the algebraic Baldwin-Lomax turbulence model is relatively computationally inexpensive and seems to provide good results, it was chosen for this simulation. Standard coefficients were used instead of the modified coefficients. A few simulations were run with the modified coefficients to verify that the overall trends would not change. Modifying the

55 44 coefficients, as suggested by [29], produced more loss in the boundary layers than previously predicted, pushing the inviscid shock structures further upstream. In essence, the modified coefficients modelled a higher back pressure of the standard coefficients, but did not change the fundamental trends of the shock structure and failure mode of the compressor. In ADPAC, wall functions are turned off on a point to point basis if the y+ spacing is less than approximately five. As shown in Table 3.3, the average values are large enough for wall functions to be utilized over most of each surface. 3.4 Solution Starting and Convergence Starting such a large numerical simulation is not trivial. The best procedure, which was determined by trial and error, started the solution using a combination of ADPAC s startup procedure and a series of small user specified perturbations made on successive solutions. Convergence of the solutions was monitored using several inviscid and viscous metrics. First, it is important to understand ADPAC s multigrid startup scheme. ADPAC s multigrid startup process initializes the domain using a user specified reference Mach number. User specified levels of coarse meshes are setup by ADPAC, where each level removes every other point in each computational direction of the previous level. The solution runs a specified number of iterations on these course mesh levels, converging the solution on a less computationally expensive mesh before increasing the density to the final mesh. After the multigrid startup, ADPAC continues to use the multigrid mesh levels to accelerate the convergence process. After a set number of iterations, ADPAC outputs a restart file, whereby the solution can be restarted from the last completed iteration. A reference Mach number of 0.15 was used to initialize each block. A low value was used to start the solution more smoothly. One-hundred iterations were run on two levels of course meshes followed by fifty on the fine mesh with an exit static pressure of 2.0 P ref. The low values helped establish a solution file with the flow vectors pointing in the correct directions.

56 45 The restart file was then used to start the solution with a slightly higher back pressure. The back pressure was raised in increments of 0.25 P ref design point. until the solution came close to the Approximately 5000 iterations were needed to fully converge the initial solution near the design point. Convergence is dictated by a drop in the RMS 5 values of three orders of magnitude and steady values of metrics such as mass flow rate, pressure ratio, and adiabatic efficiency. Some solutions did exhibit varying scales of oscillation due to unsteady phenomena. Once a converged, stable solution was obtained, all other points on the speed line were computed utilizing the restart file and small perturbations in the exit static pressure boundary condition. Different rotational speeds were also obtained utilizing the original restart. In general, 1000 iterations were needed to converge a solution. Slightly more, approximately 2000, iterations were needed to converge a solution closer to the stability limit. Figures 3.18 through 3.21 show the convergence history for one UEET case. The example is the convergence of a change in the exit static pressure boundary condition made to a fully converged solution. The example shows 1900 iterations of the multistage run. Figure 3.18 shows the mass flow rate convergence. The mass flow rate is approximately 1.5% higher than the design value. This is a result of the turbulence model. The Baldwin- Lomax turbulence model predicts a thinner boundary layer than would physically exist. Using this model, it is expected to see a slightly higher mass flow rate with ADPAC compared to the design intent or experimental verification. A difference in mass flow rate from the compressor inlet to the compressor exit can also be seen in Figure Figure 3.19 shows the exit mass flow rate non-dimensionalized with the inlet mass flow rate. The error is approximately 0.5% at the end of the convergence. The difference in mass flow rate from inlet to exit is a result of the applied boundary conditions. ADPAC is a conservative formulation, strictly adhering to mass conservation. The mixingplane and clearance patch both contribute to increasing mass flow rate through the system. 5 Error residuals are calculated as the sum of the change in the dependent variables Q, Equation 3.2. The root mean square (RMS) of the residuals is the square root of the sum of the squares of all the residuals for a block.

57 m in m out Mass Flow Rate [lbm/s] Iteration Figure 3.18: Example mass flow rate convergence history for UEET two-stage simulation 10% Nondimensional Exit Mass Flow Rate 8% 6% 4% 2% 0% -2% -4% -6% -8% -10% Iteration Figure 3.19: Example non-dimensional mass flow rate convergence history for UEET twostage simulation

58 47 The mixing-plane imposes constant circumferential values of the independent variables in the phantom cells. To meet this demand, the solution must alter in the cells preceding the boundary condition, attempting to conform to the constant circumferential value constraint, resulting in a numerical increase of the mass flow rate metric. In the clearance patch specification, cells from the suction side of the blade are patched to the pressure side of the blade and vice versa. The fluxes across the cell face area are set equal to each other. In the case of the clearance patch, the cells being patched together do not have the same cell face area. Air flows from the pressure side to the suction side through the gap. Generally, the pressure side has larger cell areas than the suction side, which results in a numerical increase in the mass flow rate across the patch. Figure 3.20 shows the log of the RMS errors. This plot clearly shows the solution is not diverging. ADPAC outputs the location of the maximum error every iteration. The noise shown in the max error plot is the result of a few cells changing consistently from iteration to iteration. The RMS error has dropped from the startup initial order of magnitude 2.0 to approximately 6.5. This is greater than a three orders of magnitude drop and is another indication of convergence and a relatively low uncertainty for this type of model. Figure 3.21 shows the convergence of the total pressure ratio and adiabatic efficiency across the compressor. Both of these metrics are relatively well behaved in this example. The viscous regions have more influence on entropy gain than the core inviscid flow. This leads to a slower convergence to such metrics as efficiency. Solutions that contain more unsteadiness also have poor convergence behavior. Oscillations can be expected, especially as the compressor nears the stall point. An average over the oscillation period is taken as a representative solution. As the model is pushed further along the compressor map with higher exit static pressure, the mass flow rate eventually begins to diverge towards zero. The integration in ADPAC fails, causing a halt to the solution iteration. The last stable, converged solution before such an occurrence is considered the stability limit for that case.

59 Log of Error Residuals RMS Error Iteration Max Error Figure 3.20: Example error residual convergence history for UEET two-stage simulation Total Pressure Ratio Adiabatic Efficiency Iteration Figure 3.21: Example total pressure ratio and adiabatic efficiency convergence history for UEET two-stage simulation

60 Solution Post-Processing This section is a brief summary of the tools used in post-processing. It is presented here only for reference. The converged solution was viewed with the commercial three-dimensional graphical viewer, FieldView. Performance characteristics were obtained using several tools created by the author for two-dimensional and one-dimensional averaging. Routines in ADPAC itself were utilized along with a few tools provided by Wellborn [24] Visualization As mentioned above, FieldView was used for all three-dimensional analysis. Figure 3.22 shows an example solution of the UEET two-stage compressor. Shock structures were visualized with contours of pressure, density, and Mach number. Weaker shocks were harder to correlate and visualize with density. In the results section, contour plots of Mach number and pressure are presented. FieldView also allows viewing of a series of solutions and the creation of flip books. This was very useful in visualizing the shock movements due to changing exit static pressure. All two-dimensional line plots were created using gnuplot 6. gnuplot is a freeware general purpose plotting program. It allows outputs to the screen and in Encapsulated Post Script, (EPS) format. Its macro language was very useful in batch plot creation Solution Averaging Post processing of a cell centered solution will inevitably involve averaging or interpolation of some kind. ADPAC outputs several relevant post-processing files. As mentioned before, a solution restart is output in cell centered format. Also, a relative and absolute reference frame Plot3D 7 file is output with the cell centered data interpolated onto the mesh. The node data is simply an average of the eight surrounding cell centered points onto the mesh 6 gnuplot c is not associated with FSF or GNU, hence the name is really gnuplot [30] 7 Plot3D is a multiple grid format. Details can be found in reference [23].

61 50 Figure 3.22: Example using FieldView for three-dimensional visualization nodes. PROBE is an ADPAC boundary condition that outputs data to a file from a specified computational plane, either mass averaged or area averaged. The Plot3D formatted files were used in three-dimensional visualization. The cell centered solution files were used for performance metrics along with the PROBE files. A FORTRAN program was written by the author to circumferentially mass and area average the cell centered solution file. In effect, this is interpolating the mesh onto the cell centered data instead of vice versa. This program was augmented with the aid of Wellborn [24] to include axisymmetric streamlines and blade surface Mach numbers. The blade surface Mach number is based on the blade surface static pressure and an upstream total pressure along a constant streamline. Contours of the blade Mach number provided an easy metric to view the position of the shock along constant axisymmetric streamlines. One-dimensional data, such as mass flow rates, pressure rise, flow coefficient, were calculated from mass averaged computational planes. Stage performance was measured from rotor inlet to stator exit. Overall performance included the IGV. It must be emphasized that this is only a model. The averaging is necessary for performance metrics. In reality, for example, the exit of stator-two has a three-dimensional density, velocity, and energy

62 51 profile. Overall absolutes of pressure ratio may not be accurate and must be compared to experimental data. However, the solutions are very precise, meaning the general trends and relative comparisons of the different solutions are valid.

63 Chapter 4 Computational Results In this section, the results of the NASA UEET two-stage compressor ADPAC numerical simulations will be presented. First, a baseline comparison to the NASA design will be discussed. Next, the stage characteristics and overall predicted performance will be presented. A detailed look at the changing shock structure at different points on the compressor map will be presented. Finally, some results showing the ultimate failure mode or the cause of the instability will be discussed. 4.1 NASA Baseline Benchmark The baseline computational case was compared to the available NASA results presented by Larosiliere [22]. It should be noted again that the two-stage design represents a refined design of the first two stages of an overall four-stage design (see Section 2.3). ADPAC did not predict the design mass flow rate at the prescribed design rotational speed (see Table 2.1), for any exit static pressure. At the design speed, rotor-one was unable to start in the ADPAC model. Solutions were gathered at various other speeds to study the swallowing capability of rotor-one. These studies resulted in the conclusion to run the compressor at 101% speed in ADPAC to study the shock structure and its relation to rotorone starting and the overall compressor stability failure mode. To verify that the 101% speed 52

64 53 Percent Span from Hub Rotor 1 NASA, 100% ADPAC, 101% Absolute Mach Number, M abs Rotor 1 Percent Span from Hub Rotor 2 NASA, 100% ADPAC, 101% Absolute Mach Number, M abs Rotor 2 Figure 4.1: Comparison of NASA design inlet profile [22] and computed axisymmetric average inlet profile from ADPAC at 101% speed for UEET rotors solution was representative of the design intent, it was compared to the through flow design results from NASA [22]. Figure 4.1 shows a comparison of the inlet Mach number profile for rotors one and two. The ADPAC data are axisymmetric mass average values of Mach number at a constant computational i plane, upstream of the rotor leading edge. The ADPAC data indicate the presence of a boundary layer at both endwalls for both rotors. Rotor-one has a subsonic region located inboard of 15% span. Traditionally, this rotor would be classified as a transonic rotor. It is considered as a supersonic rotor here, since the subsonic region seems to play no role in the choking of the rotor. Figure 4.2 shows a similar set of graphs for the UEET compressor stators. Again, the ADPAC data show the extent of the circumferentially mass averaged boundary layer thickness. The stator-one hub inlet flow appears subsonic, deviating from the mildly transonic design intent. By inspecting the three-dimensional solution, the flow vectors near the statorone hub suction side were found to be supersonic. An acceleration bubble is located at the hub on the suction side of stator-one, extending approximately 50% into the passage. Simulations suggest that stator-two has a large casing boundary layer. This is a result of the tip leakage flow of rotor-two. The large downstream influence region of the leakage flow reduces the overall circumferentially averaged momentum entering into stator-two.

65 54 Percent Span from Hub Stator 1 NASA, 100% ADPAC, 101% Absolute Mach Number, M abs Stator 1 Percent Span from Hub Stator 2 NASA, 100% ADPAC, 101% Absolute Mach Number, M abs Stator 2 Figure 4.2: Comparison of NASA design inlet profile[22] and computed axisymmetric average inlet profile from ADPAC at 101% speed for UEET stators The solution at 101% speed has nearly the same inlet flow distribution as the design intent. The design pressure ratio, efficiency, and mass flow rate can also be achieved at this rotational speed. In the following section, the stage characteristics and overall performance will be presented for both 101% and 103% rotational speeds. Running at both higher speeds served to verify that the trends were not a function of a particular rotor speed and to give further evidence of the role the shock structure plays in compressor stability. 4.2 Embedded Stage Characteristics The multistage characteristics of the UEET two-stage compressor differ from normal subsonic and transonic multistage compressor operation. A review of the subsonic operation from Cumptsy [31] is provided in Appendix C. To get a clear picture of the operation, it is useful to look at the total pressure ratio and efficiency versus corrected mass flow rate (Equation 4.1) for each stage. ṁ c = ṁ θ δ = ṁ T 0,inlet /T 0,std P 0,inlet /P 0,std (4.1)

66 55 η pc % Ω 101% Ω PR Corrected Mass Flow ( m c ) Figure 4.3: Predicted stage-two performance characteristic of UEET compressor at two rotational speeds The corrected mass flow rate is an attempt to collapse characteristics run at different inlet conditions. The inlet profile to stage-one was constant during this case study. For stage-two, however, this was particularly useful because of the changing inlet conditions to stage-two. Figure 4.3 shows the UEET compressor stage-two characteristic based on the ADPAC model. The corrected mass flow rate is evaluated from the stage-two inlet conditions. The characteristic shows a nearly vertical region followed by a bend in the curve. This bend is usually a desirable effect, resulting in a working range of mass flow rates. Generally, the stability margin is defined as the available range in the corrected mass flow rate from the design point to the stability limit. However, from the start of the bend to the stability limit, the pressure ratio is nearly constant. Very little change in the exit condition causes the stage to move along the horizontal section. This makes the range in mass flow rate less desirable, because there is no further increase in the pressure ratio.

67 56 η pc % Ω 101% Ω PR Corrected Mass Flow ( m c ) Figure 4.4: Predicted stage-one performance characteristic of UEET compressor at two rotational speeds The efficiency distributions are typical, with a peak stage polytropic efficiency of 91% at a pressure ratio of 2.048, slightly above the design pressure ratio. The bend-over in stage-two can be explained by observing the stage-one characteristic, Figure 4.4. Here, the corrected mass flow rate is evaluated from the stage-one inlet conditions. Stage-one has a vertical characteristic, usually found in the choked region of the compressor stage. The first couple of operating points are nearly identical. Stage-one starts to move along its characteristic only when the stage-two characteristic has started to bend over. A bend-over, as well as the peak efficiency of the first stage, is never reached before computed flow instabilities occur. Figure 4.5 shows the two stage overall performance characteristic. The vertical characteristic gives evidence that the compressor is always operating choked and appears to remain there even at the stability limit. To fully appreciate and understand the characteristic, it is necessary to look at the shock structure in the rotors, presented in the following section.

68 57 η pc % Ω 101% Ω PR Corrected Mass Flow ( m c ) Figure 4.5: Predicted overall performance characteristic of UEET compressor at two rotational speeds 4.3 Shock Structure Shock structure plays the key role in understanding the characteristic of multi-supersonic stage compressor performance. To highlight the important story that the shock structure tells, two sets of figures will be presented showing the UEET rotor shock structures with changing exit static pressure for the 101% speed case. Figure 2.3 in Section details the change in shock structure in a single stage for a relative supersonic airfoil section. Increasing back pressure on the compressor, or throttling the compressor, repositions the passage shock closer to the throat. The maximum efficiency point occurs when the passage shock becomes a weak shock standing at the throat. Depending on the design, the passage shock could be seen as in Figure 2.3b or even merged with the passage leg of the oblique shock. Further increase in the back pressure will pop the shock structure into a strong oblique shock detached from the blade leading edge.

69 58 Subsonic diffusion Oblique Shock Passage Shock Supersonic acceleration region Rotor 1 Rotor 2 Figure 4.6: First in series (Figures ) of Mach contour plots of the UEET compressor rotors at 50% span, 101% speed with an exit static pressure of P s,exit /P 0,inlet = 3.25 A similar trend of changing shock structure can be seen in the UEET compressor rotors. This trend can be seen in Figures 4.6 through Each successive figure represents both compressor rotors as the exit static pressure is increased. The radial plane is approximately a 50% constant span plane showing Mach number contours for rotor-one on the left and rotor-two on the right. Multiple passages are shown for clarity. The scale for all the frames is shown in the upper right corner. Red through orange represents supersonic flow. Yellow is just below the sonic point, giving an approximate location to transition. Green through blue are subsonic. With the exception of Figure 4.11b, the data shown coincide with converged solutions, each subsequent solution in sequence having a higher exit static pressure to simulate throttling of the engine. The changing exit condition is presented in the caption of each figure as a ratio of the exit static pressure at the exit plane of stator-two to the total inlet pressure taken at the inlet plane of the IGV.

70 59 Rotor 1 Rotor 2 Figure 4.7: P s,exit /P 0,inlet = 4.0 Looking at the first frame in the series, Figure 4.6, a strong oblique shock can be seen in rotor-one. Following the passage leg of the oblique shock is supersonic diffusion to subsonic velocities. A fairly weak passage shock must exist in this diffusion. Rotor-two also has a strong oblique shock. Following the passage leg of the oblique shock is a small section of supersonic acceleration aft of the throat area. A strong passage shock causes the rapid static pressure rise, reducing the flow to subsonic velocities. The apparent throat in this airfoil section is just downstream of the leading edge near the intersection of the passage leg of the oblique shock and the suction side of the airfoil. As the exit static pressure is increased, as in Figure 4.7, no discernable change occurs in rotor-one. The oblique shock of rotor-two likewise remains unchanged. The passage shock has moved forward and is becoming weaker. This figure is close to the design pressure ratio of the overall compressor.

71 60 Rotor 1 Rotor 2 Figure 4.8: P s,exit /P 0,inlet = 4.2 Rotor 1 Rotor 2 Figure 4.9: P s,exit /P 0,inlet = 4.25

72 61 Rotor 1 Rotor 2 Figure 4.10: P s,exit /P 0,inlet = 4.3 Figure 4.8 shows the solution just before the stage-two peak efficiency. What remains of the passage shock in rotor-two can still be seen. It has become much weaker and resembles the supersonic diffusion in rotor-one. The stage-two peak efficiency point in the stage-two characteristic, Figure 4.3, is shown in Figure 4.9. In this figure, the passage shock in stage-two is hardly discernable from the passage leg of the oblique shock. The passage leg of the oblique shock has increased in strength and the angle has become slightly less oblique. This shock system is the peak efficiency for stage-two, with the two shocks virtually merged together. Close observation will reveal a very minute change in rotor-one that is hardly distinguishable from numerical noise. A small increase in the exit static pressure changes the shock structure as shown in Figure The shocks in rotor-two have completely collapsed together into a strong oblique shock. The oblique shock has moved farther away from the leading edge of rotor-two and again the angle has become less oblique, meaning stronger. A small portion of subsonic acceleration

73 62 Rotor 1 Rotor 2 (a) Rotor 1 Rotor 2 (b) Figure 4.11: P s,exit /P 0,inlet = 4.32 (a) Pre rotor-one unstart (b) Post rotor-one unstart can be seen behind the oblique shock to the throat. This condition is considered to be the unstarting of rotor-two. This is the first point in the horizontal section of the stagetwo characteristic, Figure 4.3, and the first movement along the stage-one characteristic, Figure 4.4. As the exit static pressure continues to increase, the oblique shock in rotor-two becomes more normal to the flow direction, hence becoming stronger. The increasing exit static pressure can now also be felt by the upstream stage. This causes the passage shock in rotor-one to start moving forward towards the oblique shock. The last two frames in the series, Figure 4.11a and 4.11b, show the solution immediately before rotor-one unstart and immediately following rotor-one unstart for the same back pressure. Figure 4.11a shows a rotor-one shock structure similar to that of rotor-two in Figure 4.9. The passage shock is more clearly defined, but is in the process of merging with the passage leg of the oblique shock. As the solution converges, the losses in the downstream stage are growing due to the changing inlet conditions from stage-one and the growing strength of the rotor-two oblique shock. Eventually in the convergence, a dramatic change occurs when rotor-one unstarts, Figure 4.11b. The unstart of rotor-one drastically changes conditions for rotor-two. The oblique shock in both rotors is very strong and detached.

74 63 Subsonic flow following the shock in both rotors accelerates through the converging section of the rotor passage before decelerating further. The solution just before rotor-one unstart, Figure 4.11a, is the uppermost point on the performance characteristic, Figure 4.5. After rotor-one unstarts, a quasi-stable solution starts to form and the inviscid flow properties start to level off. However, eventually the solution hits a growing instability that causes the mass flow rate to drop rapidly towards zero. The solution then becomes unstable and the integration fails. It is felt by the author, that although only in a quasi-steady stage, these solutions still show the correct trend. As mentioned before, the steady model calculates a local time step based on each cell size. Smaller cells, usually in viscous regions, have smaller time steps and converge slower. Larger cells, usually in inviscid regions, have larger time steps and converge faster. The changing shock structures, being an inviscid phenomena, have the correct order and trends. Additional losses only displace the shock position further, but do not disrupt the trend. The constant radial plane shown in Figures accurately illustrates the changes occurring across the span. Understanding the changing structure in these figures helps to clarify the meridional change in the shock structure. Figures 4.12 through 4.17 are a series of figures showing the static pressure distribution on the rotor surfaces for the same set of exit static pressure conditions depicted in Figures The static pressure contours are colored with an appropriate scale for each rotor. In both cases, red represents high static pressure, or lower velocities, and blue represents low static pressures, or higher velocities. Yellow is approximately the transition region between supersonic and subsonic relative velocities. Figure 4.12 shows the static pressure distribution for the lowest point on the performance characteristic, Figure 4.5. The passage shock can be seen clearly on the pressure sides of both rotor-one and two. The suction surface shows the passage side of the oblique shock followed by the passage shock. Both rotors suction surfaces also show an acceleration bubble between the oblique and passage shocks near the tip region, indicating that the rotors are operating below the design point. The spanwise shape of the shocks in each rotor is also

75 64 Rotor 1 Rotor 2 Figure 4.12: First in series (Figures ) of static pressure contour plots on pressure and suction surfaces of the UEET compressor rotors at 101% speed with an exit static pressure P s,exit /P 0,inlet = 3.25

76 65 Rotor 1 Rotor 2 Figure 4.13: P s,exit /P 0,inlet = 4.0 different. The meridional shock shape of rotor-one has a distinctive bowed shape, while remaining perpendicular to both the hub and case. The shape of the shock in rotor-two has forward lean, similar to the shock structures found in rotors with forward sweep. It is important to understand that the airfoil shapes were adjusted in the refined design step for a tailored loading distribution. The shock structures seen in this view are the result of a design choice. Close to the overall design pressure ratio, Figure 4.13 shows, as before, no discernable change occurring in rotor-one. The rotor-two oblique shock has likewise remained unchanged. The pressure side of the passage shock in rotor-two has moved forward, with more movement occurring near the hub. The suction side has also moved forward. In all, the passage shock has become more defined from hub to tip. Figure 4.14, just before peak efficiency of stage-two, shows the rotor-two passage shock getting close to the oblique shock over most of the span. As can be seen on the rotor-two suction side, over the bottom 75% of the span the passage shock seems to have merged with

77 66 Rotor 1 Rotor 2 Figure 4.14: P s,exit /P 0,inlet = 4.2 the passage leg of the oblique shock. This seems to indicate that this part of the rotor is at or near its peak efficiency. The passage shock still exists on the pressure surface, but has grown weaker, especially over the bottom portion of the span. The tip region still has a small amount of acceleration, due to the incoming supersonic flow accelerating downstream of the throat. Figure 4.9 shows the peak efficiency point for stage-two. The suction side of the rotortwo passage leg of the oblique shock and passage shock have merged together. However, on the pressure side, the passage shock can still be seen downstream of the leading edge. The structure in rotor-one is still nearly the same. A small increase in the exit static pressure unstarts about 80% of the rotor-two span. This is illustrated in Figure The tip region on the rotor-two suction surface shows a small acceleration bubble. This indicates that the tip region is indeed still started and has not reached its local maximum efficiency point.

78 67 Rotor 1 Rotor 2 Figure 4.15: P s,exit /P 0,inlet = 4.25 Rotor 1 Rotor 2 Figure 4.16: P s,exit /P 0,inlet = 4.3

79 68 Rotor 1 Rotor 2 (a) Rotor 1 Rotor 2 (b) Figure 4.17: P s,exit /P 0,inlet = 4.32 (a) Pre rotor-one unstart (b) Post rotor-one unstart The shock in rotor-one has been displaced forward, due to the rise in exit static pressure because of the partial rotor-two unstart. This causes a change in the inlet conditions to stage-two, giving rise to the bend-over in inlet corrected mass flow rate for the stage-two characteristic, Figure 4.3. Further increase of the exit static pressure continues to move the shock in rotor-one forward. Figure 4.17a and 4.17b show the solution for pre and post rotor-one unstart. Figure 4.17a illustrates an increased bow of the rotor-one passage shock (seen on the pressure surface). As Hah [19] noticed, the shock stays perpendicular to the case and hub in the spanwise direction. Figure 4.17b shows both an unstarted rotor-one and rotor-two. As explained above, this solution is unstable. The passage shock in rotor-one pops out, merging with the oblique shock, which then detaches from the leading edge. This drastically changes the inlet conditions to stage-two and eventually causes the integration to fail. One thought is that rotor-two must remain started at the tip to increase compressor stability. This results in using such mechanisms as casing treatments or forward sweep to manipulate the shock position and structure near the tip. The loading distribution was likely tailored to constrain the passage shock in rotor-two further in the passage near the

80 69 Rotor 1 Rotor 2 Figure 4.18: Static pressure contour plot of the UEET compressor rotors at 103% speed showing a stable solution with rotor-two complete unstart, reference Figure 4.16 tip region. It was pointed out in Figure 4.16 that when rotor-two unstarted, there was a portion near the tip with a started shock structure. Solutions at 103% speed verify that this is not the case. Figure 4.18 shows a stable solution where rotor-two is completely unstarted, yet rotor-one remains started and the solution converges. This suggests the mechanism of failure is not the unstarting of the rotor-two tip region. The changing exit static pressure causes a change in the shock structure in each rotor. As seen in both the constant spanwise section and in the meridional plane, the shock in rotor-two isolates the rotor-one from higher exit static pressures. As the exit static pressure is increased, the spanwise shape of the shock only moves upstream, keeping virtually the same shape. When the downstream rotor shock unstarts or pops out of the rotor passage, the higher exit static pressures begins to affect rotor-one. The unstart of the upstream rotor only strengthens the downstream rotor unstart by decreasing the oblique angle and increasing the shock detachment distance from the leading edge.

81 Stability Limit Failure Mode One theory for compressor stability was presented in a review by Greitzer [32]. The theory states that the slope of the non-dimensional static pressure rise versus flow coefficient is zero at stall. Plotting this for each stage could reveal the limiting stage. The static pressure rise, defined by Equation 4.2, is a incompressible relation. Ψ T s = P s,stageexit P 0,stageinlet 1/2ρU 2 (4.2) As Greitzer mentioned in his review, this does not always work. Modifying the equation to reflect a compressible situation did show some helpful results. Instead of the static to total pressure rise, the static to total pressure ratio was plotted against the flow coefficient. Figure 4.19 shows this for the two stages of the UEET compressor. The axial velocity and inlet total pressure for each stage was calculated as a mass average at the inlet computational plane. The exit static pressure was calculated at the exit computational plane. It can be seen in Figure 4.19 that stage-two is approaching the stability limit as stage-one begins to move along its characteristic. The black triangles show the unstart of rotor-one. This is added to show the displacement in flow coefficient after rotor-one unstarts. discussed earlier, this is an unstable solution. The figure seems to indicate that stage-two is in fact the limiting factor. This was further reinforced by looking at stator-two before and after rotor-one unstart. Figure 4.20 shows the total pressure distribution for both stators at a constant 50% spanwise plane just before rotor-one unstart. From the low back pressure (3.25) to this point, the flow field in stator-one has remained constant. The static pressure rise has steadily increased in stator-two as stage-two moved up its characteristic. The stator-two wake can be clearly seen in the figure. Figure 4.21 shows the same solution just after rotor-one unstart. A massive separation has grown on the suction side of stator-two, due to positive incidence separation. The change in incidence angle is a result of the instantaneous drop in the mass flow rate or flow coefficient. This is depicted by the arrows in Figure 4.19 for the 101% speed case. As

82 St1: 101% Ω St1: 103% Ω St2: 101% Ω St2: 103% Ω P s,exit /P 0,inlet φ=v z /U tip Figure 4.19: Stage characteristic comparison using static to total pressure ratio from Greitzer [32] Stator 1 Stator 2 Figure 4.20: Total pressure plot of the UEET compressor stators at 50% span, 101% speed before rotor-one unstart

83 72 Stator 1 Stator 2 Figure 4.21: Total pressure plot of the UEET compressor stators at 50% span, 101% speed after rotor-one unstart To verify nothing unexpected is happening in the flow field near the rotor-two or statortwo trailing edge, Figures 4.22 and 4.23 are shown. Figure 4.22 shows a constant i (axial direction) computational plane near the rotor-two trailing edge. The leakage jet from the rotor-two casing clearance is very distinct. The influence region is also fairly large, aiding in endwall separation at the case. There is also a suction side separation bubble from the corner vortex near the rotor-two hub. These structures only seem to contribute in modifying the shock structure position. The leakage vortex influence region did move forward and become larger near stall, but did not cause an instability by jumping ahead of the leading edge. Figure 4.23 shows a constant i computational plane near the stator-two trailing edge. The notable features include the hub clearance vortex, the corner vortex at the case, and the suction side separation. This solution is before rotor-one unstart and shows a separation area mostly in the mid span region. This separation grows, as shown in Figures 4.20 and 4.21, ultimately leading to compressor stall.

84 73 Case R2 TE Leakage Flow Jet Separation Hub Rotation Figure 4.22: Rotor-two constant axial direction plane near trailing edge Case S2 TE Corner Vortex SS Separation Leakage Flow Vortex Hub Rotation Figure 4.23: Stator-two constant axial direction plane near trailing edge