https://ntrs.nasa.gov/search.jsp?R=19810016540 2019-04-06T13:47:56+00:00Z NASA CR 3426 c. 1 NASA Contractor Report 3426 Semi-Actuator Disk Theory for Compressor Choke Flutter J. Micklow and J. Jeffers CONTRACT NAS3-20060 JUNE 1981 TECH LIBRARY KAFB. NM NASA Contractor Report 3426 Semi-Actuator Disk Theory for Compressor Choke Flutter J. Micklow and J. Jeffers Umited Techologies Corporation West Palm Beach, Florida Prepared for Lewis Research Center under Contract NAS3-20060 NASA National Aeronautics and Space Administration Scientific and Technical Information Branch 1981 TABLE OF CONTENTS SUMMARY. .......................................................... iv INTRODUCTION .................................................... 1 ANALYTICAL MODEL ................................................ 3 Model Definition ................................................ 3 Assumptions and Boundary Conditions ............................. 3 Assumptions .................................................. 3 Boundary Conditions .......................................... 4 Derivation of the Unsteady Model ................................ 5 Upstream and Downstream Irrotational Flow Equations .......... 5 Intrablade Flow Equations and Solutions ......................... 9 Perturbation Equations and Solutions for Region l............ 9 Perturbation Equations for Region 2 .......................... 14 Unsteady Shock Movement ...................................... 17 Perturbation Equations for Region 3 .......................... 19 Rotational Downstream Flow Field ............................. 20 Continuity and Momentum Equations for Region 3 ............... 21 RESULTS .......................................................... 22 Wind Tunnel Test Data ........................................... 22 Flutter Analysis ................................................ 22 Computational Method ......................................... 22 Flutter Prediction Results ................................... 24 Summary of Results ........................................... 29 Conclusions .................................................. 30 APPENDIXES A DERIVATION OF THE SMALL PERTURBATION FORM OF THE EQUATIONS OF MOTION MOTION IN REGION l.................................... 31 B SOLUTIONS TO THE EQUATIONS OF MOTION FOR REGION l............... 39 C SOLUTIONS TO THE EQUATIONS OF MOTION FOR REGION 2 ............... 63 D SOLUTIONS TO THE EQUATIONS OF MOTION FOR REGION 3 ............... 74 E STEADY-STATE FLOW COEFFICIENTS FOR REGIONS 1, 2 AND 3 ........... 82 F CALCULATION OF MEAN FLOW AERODYNAMICS.. ......................... 98 G DEFINITION OF AREA PERTURBATIONS.. .............................. 100 H CALCULATION OF THE STEADY-STATE SHOCK LOCATION AND TEST FOR CHOKED FLOW.. ................................................... 109 I LIFT AND MOMENTC OEFFICIENT CALCULATION. ........................ 112 J COMPUTERC ODE COMPILATION ....................................... 114 REFERENCES ....................................................... 160 NOMENCLATURE.. .................................................. 162 iii SUMMARY Utilizing semi-actuator disk theory, a mathematical analysis was developed to predict the unsteady aerodynamic environment for a cascade of airfoils harmonically oscillating in choked flow. In the model, a normal shock is located in the blade passage, its position depending on the time dependent geometry and pressure perturbations of the system. In addition to shock dynamics, the model includes the effect of compressibility, interblade phase lag, and an unsteady flow field upstream and downstream of the cascade. Calculated unsteady aerodynamic forces using the semiactuator disk model were com- pared to experimental data from isolated airfoil wind tunnel tests. The wind tunnel data simulate the special cascade condition of 180 deg interblade phase. Agreement between experimental and theory was reasonable. The semiactuator theory was also evaluated using compressor airfoil choke flutter data from single-spool tests of the FlOO turbofan engine. The model was incorporated into a flutter prediction program in which calculated aerodynamic damping is correlated to construct flutter onset boundaries. The calculated flutter boundaries compared well with the measured flutter boundaries. Based on these evaluations, it was concluded that a conservative choke flutter design system could be established based on the semiactuator disk model. INTRODUCTION Compressor airfoil flutter remains a continuing problem in the design and development of advanced aircraft gas turbine engines. Flutter occurs over a wide range of operating condi- tions, but can be categorized into four regions: (1) subsonic/transonic stall, (2) subsonic/tran- sonic choke, (3) supersonic unstalled, and (4) supersonic stalled, as shown in Figure 1. Operating Stalled Line Supersonic Flutter - I Surge Line Subsonic \i ii \+ Stall Flutter- Unstalled 1 Supersonic Flutter- 1 Weight Flow FD1 97986 Figure 1. Possible Flutter Boundaries The subsonic stall flutter problem has been investigated by a number of authors. Jeffers (Reference 1) devised a semi-empirical unsteady aerodynamic theory based on combining the unsteady unstalled aerodynamic forces from Smith’s theory (Reference 2) with correction from theory and experimental data of isolated airfoils operating at high incidence in incompressi- ble flows. Sisto (Reference 3) used steady aerodynamic data to treat the unsteady flow prob- lem in a quasi-steady manner. Perumal (Reference 4) developed an essentially “Helmholtz flow” model, while Yashima and Tanaka (Reference 5) adapted a rigid wake model to obtain reasonably good correlation with linear cascade experimental data. Most recently, Chi (Refer- ence 6) used a small perturbation technique to model flow separation. The supersonic flow region has also been discussed by a number of authors. For the unstalled regime, a finite difference method was first used by Verdon (Reference 7) and Brix and Platzer (Reference 8) to model the unsteady supersonic aerodynamics. Other approaches by Kurosaka (Reference 9) and Verdon and McCune (Reference 10) extend a velocity potential method first developed by Miles (References 11 and 12) for simple supersonic cascade configurations. I; I Recently, an unsteady actuator disk model was developed by Adamczyk (Reference 13) with encouraging results for supersonic stall bending flutter. The supersonic stalled region was also investigated by Goldstein, Braun and Adamczyk (Reference 14) in which the small perturbation analysis included the presence of a strong in passage shock. The choke flutter problem that has arisen in advanced gas turbine engines with variable inlet guide vanes poses a very serious problem and no analytical model exists at present to predict the unsteady aerodynamic environment. The complex nature of this environment has thus far resisted rigorous mathematical formulation, but a “simplified” model has been under- taken herein based on a modified semi-actuator disk approach with one-dimensional channel flow. The channel flow approach originally used by NASA-NACA to analyze inlet diffusers of ramjet and turbojet engines was selected because airfoil cascades can exhibit flow characteris- tics similar to those of inlet diffusers. The flow in an inlet diffuser and a choked blade pas- sage both contain a shock wave whose position strongly affects the pressure forces on the channel walls or blade surfaces. The position of the shock depends upon channel geometry and, therefore, in the case of the airfoil cascade, can be related to the vibratory motion of the airfoils in flutter. A preliminary analysis was completed in the initial phase of this effort which produced promising results when used in a stability prediction of a compressor rotor that experienced choke flutter at off-schedule operating conditions. However, concern for cer- tain aspects of the preliminary model led to the present approach which includes a modified semi-actuator disk method to describe the upstream and downstream flow fields. The section following contains the analytical derivation and definition of the mathemat- ical model, including a steady-state interblade analysis and a linearized small perturbation analysis. The next section details the results obtained using this channel flow model. In order to rid the text of this report of complex, cumbersome, and lengthy mathematical manipula- tions and assumptions, numerous appendices are included herewith which allow model devel- opment in a straight forward manner. r ANALYTICAL MODEL Model Definition The semi-actuator disk model consists of two solutions: a steady-state intrablade analysis and an unsteady linearized small perturbation analysis. The steady-state analysis utilizes steady isentropic one-dimensional relations to define the intrablade conditions. An iterative procedure, ending in the match of the known static pressure ratio across the blade, locates the steady-state normal shock position. The procedure appears in Appendix H. The flow entering and leaving the cascade is defined externally by a streamline analysis. The unsteady solution consists of three basic flow fields: (1) upstream flow field, (2) intrablade flow analysis, and (3) a downstream flow field, as shown in Figure 2. e-Dimensional, Inviscid. nsteady, Compressible Flo Two-Dimensional, tnviscid. Rotational, Unsteady Compressible Flow Two-Dimensional, Inviscid lrrotational, Unsteady Compressible Flow Figure 2. Unsteady Flow Field Description Assumptions and Boundary Conditions Assumptions The following assumptions were made relative to the flow within the three basic flow fields of the unsteady solution: 1. Upstream Flow Field - The flow is assumed to be two-dimensional, inviscid, irrotational, unsteady, and compressible. 3 2. Intrablade Flow Analysis - The flow is assumed to be one-dimensional, inviscid, unsteady, and compressible. Figure 3 details the division of the flow field into three sections: a subsonic section from blade leading edge to the blade throat or M = 1, a supersonic section from blade throat to shock location, and a subsonic section from shock location to blade trail- ing edge. 3. Downstream Flow Field - The flow is assumed to be two-dimensional, rotational, inviscid, compressible, and constructed of the sum of two basic solutions: an irrotational part similar to the upstream flow field and a rotational part. due to the vortices being shed off the blade trailing edge. ‘2 Exit Flow Location d Angle M-Cl M>l / \ \ % Inlet Flow Angle Stagger Angle \ Figure 3. Intrablade Flow Field Boundary Conditions Boundary conditions for the unsteady solutions consist of the following: 1. The mass flow is continuous at the leading- and trailing-edge lines. 2. Conservation of mass, energy, and momentum was observed within each section of the blade channel. 3. The Kutta condition at the trailing edge is satisfied by specifying the exit air angle. 4