Mixing-Plane Method for Flutter Computation in Multi-stage Turbomachines

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA 2009-862 Mixing-Plane Method for Flutter Computation in Multi-stage Turbomachines Roy Culver, and Feng Liu The mixing-plane method for calculating the three-dimensional flow through multistage turbomachinery is used to perform flutter analysis on a single stage transonic compressor. The turbomachine considered is composed of an inlet guide vane (IGV) and a compressor blade (NASA Rotor 67). The mixing-plane boundary condition enables steady and unsteady computation of the flow through multiple blade rows. This allows incorporation of multi-stage effects without having to perform computationally intensive fully unsteady multi-stage flow computations. Forced motion flutter computations are performed for both the isolated compressor and with the IGV. Comparison between the damping ratio shows a decrease in stability when the IGV is included in the computation. I. Introduction The modern desire to make turbomachinery blade rows both lighter and better performing pushes the designs to the extreme. As blade rows are designed to be lighter they also become less rigid. Also, as the blade rows are made to operate under more sever conditions the blades are put under more stress. One negative consequence of these design trends is that blade rows may become more susceptible to flow induced vibrations. These vibrations may fatigue the blades and/or lead to catastrophic failure. If this fate is to be avoided, the designer must be aware of the possible sources of the vibrations, and the mechanisms involved. Because turbomachines are highly dynamic, there are many ways in which harmful vibrations may be internally generated. One important source of these vibrations is the unsteady flow field that results from the interaction of consecutive blade rows in multi-stage turbomachines. The practice of using CFD solvers to obtain steady flow solutions for three-dimensional turbomachinery blade rows has been maturing for about twenty years. While these methods may not be in standard use for industrial design yet, they have certainly been relied upon more and more in recent years. However, a shortfall of these solvers is that they are unable to account for the effects arising from the interaction of multiple blade rows. Since advanced turbomachines are typically comprised of many closely spaced stages, these effects should be considered as they play an important role in flow field, and thus may be critical to understanding the response of blade rows to flow induced vibrations. In response to this need for understanding, work has been done in the area of fully unsteady flow solutions to consider blade row interactions. 1 These types of methods are able to resolve the complete unsteady flow field and may give the designer a high fidelity view into the flow domain. Unfortunately, these methods are highly expensive computationally and as such are not practical for a designer who needs performance predictions in a matter of hours, not days. In attempts to find somewhat more practical methods for considering the flow through multiple blade rows, several researchers have worked on approximate methods. One such method approximates the unsteady flow field that occurs in multi-stage turbomachines with a steady flow by including a mixing layer between adjacent blade rows. The idea is that by adding this mixing layer between adjacent blade rows, the flow can essentially mix out the circumferential variations, which are the cause of some of the unsteadiness in a real multi-stage machine. While this method smooths out the circumferential variations, it still retains variations in the axial and radial directions, enabling the researcher to examine how these variations effect the multistage flow field. Graduate Student Researcher, Mechanical & Aerospace Engineering Department, University of California Irvine Professor, Mechanical & Aerospace Engineering Department, University of California Irvine, Associate AIAA-Fellow 1 of 8 Copyright 2009 by the American Institute of Aeronautics and American Astronautics, Institute Inc. All of rights Aeronautics reserved. and Astronautics

In 1992 work by Denton 2 demonstrated an approach to this method he called an Interrow Mixing Model. In that work he coupled an Euler Solver with various methods of approximate solution for viscous stresses. Variations in the circumferential direction were also accounted for at the mixing layer boundary in order to eliminate improper loading on the leading and trailing edges of the blades. This early work demonstrated that results for the steady flow through multi-stage turbomachines could be obtained by this approach. Later work by Chima 3 used this approach to compute the flow through the space shuttle main engine fuel turbine. His work examined several different averaging methods as well as interface boundary conditions in order to examine their effects on solution accuracy. Significant was his inclusion of a 1-D characteristic boundary condition 4 and the mixing-plane interface which help eliminate distortions in the pressure field which result from a more commonly used constant pressure exit condition. Recent work by Davis 5 considered the flow through the NASA Stage 35 single stage compressor using a mixing-plane approach for both steady and unsteady flows. Here the focus was on predicting the flow over a wide range of operating conditions (choke to stall) for a fixed speed. By modeling the flow physics more closely than isolated blade row studies, these simulations were able to more clearly identify the unsteady mechanisms responsible for the onset of stall in the machine. While there has been considerable work done to resolve the flow field in multi-stage turbomachines, there has been relatively little work done to examine the interaction of this flow with the structural dynamics of turbomachines. One example of this is the recent work by Vahdati, Sayma and Imregun. 6,7 Here, free vibration computations were performed for several three dimensional multi-stage test cases. A viscous flow solver was coupled with a linear, modal superposition, structural solver to simulate the aeroelastic system. As a testament to the fidelity of this type of model, a good comparison was made to the available experimental data for the maximum tip displacement observed under self excited vibration. One point which is emphasized by the authors is that the low frequency unsteadiness arising from the flow is much less understood that the deterministic unsteadiness linked to the shaft rotation rate. The current work is motivated both by the desire to improve the understanding of multi-stage turbomachine flows and the desire to analyze the fluid-structure interaction present in such machines. To that end, a three-dimensional flow solver is to obtain the steady flow solution for a multi-stage turbomachine using the mixing-plane approach described above. Proceeding on from this steady result, flutter analysis is conducted. Here only forced motion cases are performed. Isolated blade row forced motion computations are also performed and the results are compared to determine what effects, if any, the incorporation of multiple blade rows has on the computed flutter sensitivity of a transonic compressor fan blade row. II.A. Flow Solver II. Computational Method A density-based finite-volume method is used to solve the conservative laws for unsteady compressible flow. The Favre-averaged Navier-Stokes equations may be written as WdV + F nds = 0 t V where V is an arbitrary control volume with boundary surface S, and n is the unit normal to the surface (directed outward). The state vector W is ρ W = ρu ρe where ρ is density, u is the three dimensional velocity vector, E is the energy. The Flux vector F = F i F v may be split into is inviscid and viscous components F i = ρu T ρuu + p ρeu + pu T, F v = S 0 τ (τ u q + (µ + σ ) T k) T 2 of 8

where τ is the shear stress tensor and q is the heat flux vector. The flow solver used to solve these equations is based on the code ParCAE which was developed and used for the study of three-dimensional turbomachinery aeroelasticity. 8 ParCAE is an explicit, multistage, multiblock, parallel flow solver which uses dual time stepping to resolve unsteady flows. An implementation of the Spalart-Almaras turbulence model has been used for all viscous simulations shown in this work. 9 II.B. Structural Solver The structural model Incorporated into ParCAE is a modal superposition method. Starting from the structural equations of motion for a system with a finite number of degrees of freedom M q + C q + Kq = F where q is the displacement vector. Using a modal approach, the solution has the form q = N η i Φ i i=1 where Φ i are the eigenmodes and η i are the generalized displacements. If we diagonalize this system using the matrix of eigenvectors Φ, we obtain an expression for the generalized displacement of the structure η i + 2ζ i ω i η i + ω 2 i η i = Q i Where the aerodynamic forces F are projected onto the mode shapes to obtain the generalized forces Q i, for each mode. Here, ω i are the natural frequencies, and ζ i are the damping coefficients for each mode. When solving the structural equations with ParCAE, the modes shapes for the structure are generated for each geometry as a preprocessing step and may be obtained from either a finite element solver or from experimental measurement. An explicit method is used to integrate the structural equations of motions in time. An efficient transfinite interpolation method is used to interpolate structural deformations across the entire grid after each time step, and thus avoid completely re-meshing the domain. II.C. Boundary Conditions II.C.1. Inlet/Outlet Boundary Conditions For the flows considered here, a radial distribution of total pressure, total temperature and flow angles is specified while one characteristic is extrapolated for the inlet boundary condition. At the outlet, the static pressure is specified while the remaining four characteristics are extrapolated. II.C.2. Interface Boundary Condition At the mixing-plane interfaces, radial profiles of flow variables are specified as the initial condition. While the code is running, updated profiles are obtained using mixed-out averaged flow variables 3 in the first interior cell each block adjacent to the mixing-plane. These interior profiles are then passed to the ghost cells on the opposite side of the interface. As a result, when a converged solution is obtained, the circumferentially averaged flow variables at the mixing plane will be equivalent on either side of the interface and thus mass, momentum and energy will be conserved. However, as pointed out by Denton, 2 entropy will be generated by this mixing process. A diagram of the use of ghost cells for this mixing-plane boundary condition is included as Figure II.C.2. In order to compute the mixed-out flow variables, the expressions given in Equation 1 are used. Here, a local coordinate frame which is aligned with the mixing-plane has been used to write the inviscid fluxes through the interface. The subscript n refers to direction normal to the surface. The subscript t refers to 3 of 8

Mixingplane Interface w(r) Upstream Block Downstream Block w(r) Figure 1. Diagram of the update procedure at the mixing-plane interface the radial direction, and θ refers to the circumferential direction, which is also tangent to the mixing-plane. I 1 I 2 I 3 I 4 I 5 where the expression for specific total energy E is ρ U da = ρ U n [ρu n U + p e x ] da = ρ U n U n + p [ρu θ U + p e θ ] da = ρ U θ U n [ρu t U + p e t ] da = ρ U r U n [ρ E + p]u da = ρ E U n + p U n E = 1 p γ 1 ρ + 1 ( U 2 2 n + Uθ 2 + ) U2 t These expressions taken together represent a quadratic equation for pressure, who s solution is p = 1 [ ] I 2 ± I2 2 γ + 1 + (γ2 1)(c 2I 1 I 5 ) where and the remaining flow variables are determined as II.D. Flutter Analysis c = I 2 2 + I 2 3 + I 2 4 U n = I 2 p I 1, U θ = I 3 I 1, U t = I 4 I 1, ρ = I2 1 I 2 p The flutter analysis performed here consists of so called forced motion tests. Forced motion tests are conducted by applying a prescribed motion to the structure in question and observing the resulting flow field. (1) 4 of 8

(a) Top View (b) Side View Y Z X (c) Isometric View Figure 2. Rotor 67 with IGV configuration In this case, no structural dynamics model is needed as the motion is prescribed. The primary result for this type of test is the damping ratio Ξ, which is defined as Z T 1 Cη dη (2) Ξ= 2 πh 0 where Cη is the force coefficient aligned with the generalized displacement η. and h is the normalized magnitude of oscillation. The damping ratio provides a measure of the aerodynamic damping on the structure. A negative damping ratio indicates that work is being done on the structure, and thus the condition may be classified as unstable. III. Results ParCAE has been previously demonstrated for computation of 3D turbomachinery flutter8, 10 in isolated blade rows. To demonstrate ParCAE s ability to perform multi-stage computations, NASA s Rotor 67 transonic compressor blade row has been used.11, 12 Although the NASA Rotor 67 case was designed to operate in an isolated configuration, for the purposes of this study, an artificial inlet guide vane (IGV) was created using airfoils from the NACA four digit airfoil series which varied in thickness from 6 percent at the hub to 12 percent at the shroud. Although the purpose of an IGV is typically to align the inlet flow when a compressor is operating in an off-design condition, the IGV is kept parallel to the flow for this case. Top, side and isometric views of the geometry are given in Figure 2. For the forced-motion computations, a fictitious torsional mode shape was used as torsional modes tend to be the most sensitive to vibrations for highly twisted transonic compressor blades. Steady flow computations were performed for both the isolated Rotor 67 case as well as the case with the IGV. The performance line for the isolated computations is compared with experimental data in Figure 3. Notice that near the choke and the maximum efficiency conditions, the agreement with experiment is quite good. This agreement breaks down in the near stall condition. This may be due in part to the over prediction 5 of 8

0.92 0.9 0.88 η 0.86 0.84 0.82 (a) Total Pressure Ratio 1.7 1.65 1.6 P 02 /P 01 1.55 1.5 1.45 experiment current 1.4 1.35 1.3 0.94 0.96 0.98 1 m dot /m dot,choke (b) Adiabatic Efficiency Figure 3. Performance From Steady Computations for the Rotor 67 Compressor of tip leakage flow, which may be exaggerated at higher pressure ratios. To give a more detailed description of the flow field, circumferentially averaged radial profiles of flow variables were computed at the inlet and outlet of the Rotor domain for a near-stall operating condition. These are shown in Figure 4. The agreement for both the flow angle and Total temperature ratio are excellent for this condition. The largest discrepancy is seen in the downstream position from mid-span to the tip for the Total pressure ratio. This coincides with the under predicted total pressure ratio seen in Figure 3. Even with the under prediction of the pressure ratio for the near stall conditions, the trends in the performance are as expected and the agreement of the radial profiles is good. That being the case, the results of forced-motion computations may still be instructive as to the effect of multiple blade rows on flutter behavior. These computations were performed for both the case with and without the IGV at the maximum efficiency point (about 99% of the choked mass flow rate). The results are shown below in Figure 5 and Table 1. Figure 5 shoes the time history of the modal component of the resultant aerodynamic force on the deforming Rotor 67 blade for the last several periods of the unsteady computation. Here the green line is for the isolated rotor case and the blue line is for the case with the IGV. Also plotted is the prescribed sinusoidal magnitude of the modal deformation. Note that the magnitude of the forces are significantly larger for the case without the IGV. Using Equation 2 we are able to compute damping coefficients for both cases. Table 1 shows the results of this computation. The case with the IGV shows a significant decrease in the damping coefficient, representing a decrease in stability of approximately 70%. Table 1. Damping Coefficient for Forced-Motion Case Case ζ Isolated 0.53579 With IGV 0.16046 6 of 8

0.25 Experiment downstream ParCAE downstream Experiment upstream ParCAE upstream 0.2 r 0.15 0.1 1 1.2 1.4 1.6 P 02 /P 01 (a) Total Pressure Ratio 0.25 0.25 0.2 0.2 r r 0.15 0.15 0.1 1 1.05 1.1 1.15 1.2 1.25 1.3 T 02 /T 01 (b) Total Temperature Ratio 0.1 0 20 40 60 80 Beta (c) Flow Angle Figure 4. Radial profiles of flow variables in the near stall flow condition IV. Conclusions and Future Work In this work, the mixing-plane method has been used to perform steady and unsteady computations of the flow through a single stage transonic compressor comprised of a fictitious IGV and the NASA Rotor 67 blade row. In the steady computations, an under prediction of performance was observed. This has been at least partially attributed to the over prediction of the tip leakage flow in the high pressure flow conditions. The circumferentially averaged radial profile of total pressure also show this discrepancy at the near stall condition from mid-span to the tip. However, the radial profiles of total temperature and flow variables were in very good agreement with the experiment. The results of the forced-motion computations showed a significant decrease in stability when the fictitious IGV was included in the computational domain. Even though these computations are very limited in scope, their implications are significant. The results indicate that multi-stage effects may play a large role in the aeroelastic stability of transonic compressors such as NASA s Rotor 67. It also significant that this was found even when ignoring the unsteady circumferential disturbances which are ignored by the mixing-plane method. If nothing else, this results gives motivation for further studies. The authors intend to extend this work in several ways. First, by performing these forced-motion comptutation over a wide range of inter blade phase angles and operational conditions, a more complete picture 7 of 8

0.04 eta force isolated force with igv 1.5 1 F-F0 0.02 0-0.02 0.5 0-0.5 Modal Displacement (eta) -1-0.04-1.5 0 1 2 3 4 5 Time Figure 5. of the stability domain for this configuration will be obtained. Secondly, by performing free-response computations for the cases with and without the IGV, we will examine what indications the free response method may give as to multi-stage effects on the flutter sensitivity of transonic turbomachines. Also, a study of the effect of IGV spacing on the stability of these blade rows will be performed. Finally, fully unsteady computations are intended for the configurations described in this paper. Although, these computations are not currently practical for design purposes, they may provide useful insight into fully unsteady flow field, which is not captured by the mixing-plane computations. References 1 Rai, M. M., Unsteady Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-Stator Interaction, Journal of Propulsion and Power, Vol. 5, No. 3, May-June 1989, pp. 307 319. 2 Denton, J. D., The calculation of three dimensional viscous flow through multistage turbomachines, Transactions of the ASME, Vol. 114, January 1992, pp. 18 26. 3 Chima, R. V., Calculation of Multistage Turbomachinery Using Steady Characteristic Boundary Conditions, Tech. Rep. NASA-TM-1998-206613, NASA, 1998. 4 Giles, M. B., Non-Reflecting Boundary Conditions for Euler Equation Calculations, AIAA Journal, Vol. 28, No. 12, 1990, pp. 2050 2058. 5 Davis, R. L. and Yao, J., Prediction of Compressor Stage Performance from Choke Through Stall, Journal of Propulsion and Power, 2006. 6 Sayma, A. I., Vahdati, M., and Imregun, M., An Integrated Nonlinear Approach For Turbomachinery Forced Response Prediction. Part I: Formulation, Journal of Fluids and Structures, Vol. 14, 2000, pp. 87 101. 7 Vahdati, M., Sayma, A. I., and Imregun, M., An Integrated Nonlinear Approach For Turbomachinery Forced Response Prediction. Part II: Case Studies, Journal of Fluids and Structures, Vol. 14, 2000, pp. 103 125. 8 Sadeghi, M., Parallel Computation of Three-Dimensional Aeroelastic Fluid-Structure Interaction, Ph.D. thesis, University of California, Irvine, 2004. 9 Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries, 1998. 10 Sadeghi, M. and Liu, F., Coupled Fluid-Structure Simulation for Turbomachinery Blade Rows, No. AIAA Paper 2005-0018, Reno, NV, Jan 2005. 11 Reid, L. and Moore, R. D., Performance of a Single-Stage Axial-Flow Transonic Compressor with Rotor and Stator Aspect Ratios of 1.19 and 1.26, Respectively, and with Design Pressure Ratio of 1.82, Tech. Rep. NASA-TP-1978-1338, NASA, 1978. 12 Reid, L. and Moore, R. D., Deisng and Overall Performance of Four Highly Loaded, High-Speed Inlet Stages for an Advanced High-Pressure-Ratio Core Compressor, Tech. Rep. NASA-TP-1978-1337, NASA, 1978. 8 of 8