Computation of cascade flutter by uncoupled and coupled methods

Transcription

1 International Journal of Computational Fluid Dynamics, Vol. 19, No. 8, November 25, Computation of cascade flutter by uncoupled and coupled methods MANI SADEGHI and FENG LIU* Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA , USA (Received 23 April 25; revised 26 October 25; in final form 26 October 25) 1. Introduction A computational method for flutter prediction of turbomachinery cascades is presented. The flow through multiple blade passages is calculated using a time-domain approach with coupled aerodynamic and structural models. The unsteady Euler/Navier-Stokes equations are solved in quasi-threedimensions using a second-order implicit scheme with dual time-stepping and a multigrid method. A structural model for the blades with bending and torsion degrees of freedom is integrated in time together with the flow field. Information between structural and aerodynamic models is exchanged until convergence in each real-time step. Computational results for a cascade are presented and compared with those obtained by the conventional energy method and with experimental and numerical data by other authors. Significant differences are found between the coupled and uncoupled methods at low mass ratios. A transonic test case with strong nonlinear phenomena is investigated with the fluid structure coupled method. Results for inviscid flow are compared with results of Navier-Stokes computations. Keywords: Cascade; Flutter; Fluid structure interaction; Navier-Stokes equations General aeroelastic analyses are concerned with the effects of fluid flow on the structure it surrounds and vice versa. The flow and the structure behave as a single aeroelastic system, despite being separated by a material boundary. Flow and structure impose boundary conditions on each other. The boundary of the flow domain is given by the structural surface, whose shape and motion is determined by the structural dynamics. In turn, the structural dynamics are influenced by aerodynamic forcing. Additional coupling between flow and structural dynamics may be caused by exchange of thermal energy. The coupling between flow and structure leads to a behavior generally termed fluid structure interaction. Although always present, coupling between flow and structure is not always strong. For example, in turbomachinery flutter, the effect of the flow on the oscillation frequency is conventionally considered negligible. This is based on the assumption of relatively small force perturbations acting on blades with relatively large inertia and stiffness. Consequently, flutter stability is determined by solving the unsteady flow over blades subject to prescribed oscillation, where frequency and mode shapes are usually chosen to be the same as in free vibration. Neighboring blades are assumed to oscillate harmonically at the same interblade phase angle throughout the cascade by use of the traveling wave model of Lane (1956) for tuned cascades. The work that the aerodynamic forces perform on the oscillating blade serves as the measure for instability (Carta 1967). The assumption of small fluid structure coupling is often adequate, when concerned with the onset of flutter. It breaks down only for systems with small inertia, i.e. with light blades. Furthermore, nonlinear flutter behavior, such as limit-cycle oscillation, is not predictable without solving the full aeroelastic system of equations, including the structural equations. Although not common in turbomachinery, cases of nonlinear flutter have been demonstrated numerically by Carstens and Belz (21). Moreover, even small coupling effects may be significant in the study of other small parameters such as mistuning. Using a fluid structure coupled approach, Sadeghi and Liu (22) show that frequency mistuning affects the flutter stability only if the difference of natural frequencies of neighboring blades is above a finite minimum. If the mistuning is too small, e.g. within random variations due to manufacture tolerances, the blades still lock into a mutual *Corresponding author. fliu@uci.edu International Journal of Computational Fluid Dynamics ISSN print/issn online q 25 Taylor & Francis DOI: 1.18/

2 56 M. Sadeghi and F. Liu oscillation frequency. This behavior is attributed to fluid structure interaction in combination with blade-to-blade aerodynamic coupling. With focus on small oscillations, the flow solution is often sought by a linear method. It is assumed that the flow can be described by a superposition of a nonlinear steadystate and a small perturbation which is subject to the linearized flow equations. This approach is attractive because the solution of the linearized equations requires much less computational effort than a time-marching solution of the unsteady equations. By applying frequency-domain methods, the periodic solution to a prescribed harmonic motion is sought directly, without the need to resolve transient behavior. Hall and Clark (1993a, b) apply linearized Euler computations to solve for the unsteady flow in oscillating two-dimensional cascades under subsonic and transonic conditions. The approach is shown to accurately predict shock impulses if the applied linearized finite-volume scheme is conservative. A time-linearized method has also been applied to the Navier-Stokes equations, in a study of dynamic stall on oscillating blade profiles by Clark and Hall (2). However, under transonic flow conditions, even seemingly small oscillations may lead to nonlinear flow behavior. Huff et al. (1991) solve the nonlinear unsteady Euler equations on oscillating cascades with a timemarching method. They investigate amplitude effects on the unsteady pressure over the harmonically pitching blade in a cascade. The flow is considered linear if the amplitude of the unsteady pressure is a linear function of the pitching amplitude. In the case of transonic flow with a strong shock, nonlinearity is shown to occur at a pitching amplitude of about 18, depending on the applied interblade phase angle. At an interblade phase angle of 188 the nonlinearity appears at even lower pitching amplitudes. The oscillating shock is shown to intermittently choke the cascade, thus resulting in a flow that fundamentally differs from the steady-state solution. Moffatt and He (23) present an efficient frequencydomain method for predicting the forced response of turbomachinery blade rows. In a novel energy method, the forcing work is assumed to increase linearly and the damping work to increase quadratically with the blade vibration amplitude. This is consistent with the assumption that the unsteady flow is subject to the linear equations. The steady-state vibration amplitude is then sought by balancing the calculated amplification and damping. The description of the unsteady flow perturbation can be further simplified by applying a reduced-order model (ROM), which facilitates parametric studies. Hall et al. (1995) perform an eigen analysis of the linearized twodimensional potential equations to construct a ROM for the flow perturbation on oscillating blades. With a specified interblade phase angle, the ROM is applicable for a range of oscillation frequencies and arbitrary mode shapes. A proper orthogonal decomposition method is applied by Epureanu et al. (2), to create a ROM for transonic viscous flow. The unsteady flow on oscillating cascades is described by coupling the potential equations with an integral boundary layer model. While the above linear methods are particularly useful for parametric studies in preliminary design stages, full account for aeroelastic behavior in general situations requires the simultaneous solution of the dynamic structural and flow equations. As the performance of computer hardware increases, more complex analyses become feasible and allow for more realistic solutions. Instead of applying small-disturbance theory, the unsteady Navier-Stokes equations and the structural equations are solved simultaneously. The solution is most commonly obtained by a time-marching method. Sisto et al. (1991, 1993) apply a coupled method to study stall flutter in a linear cascade. They use a vortex and boundary-layer method for incompressible flow coupled with a spring model for the blade motion. A similar torsionalspring and linear-spring model for rigid profiles is used by Bakhle et al. (1992), and Alonso and Jameson (1994), Hwang and Fang (1999), Bakhle et al. investigate potential flow through a cascade. Hwang and Fang solve the Euler and Navier-Stokes equations through a cascade including a transonic test case and a case of stall flutter. Alonso and Jameson perform parallel computation of wing flutter. Ji and Liu (1999) developed a Navier-Stokes code to calculate quasi-three-dimensional unsteady flows around multiple oscillating turbine blades. A time-accurate multigrid method is used applying the pseudo-time approach by Jameson (1991). Computations on multiple blade passages are performed in parallel by using the message passing interface (MPI) without the use of phaseshifted periodic boundary conditions. Therefore, the blade motion does not have to be periodic, neither spatially nor temporally. Taking advantage of this property, the present authors used this code to study flutter of blades mistuned either in frequency or phase with an uncoupled approach (Sadeghi and Liu 21). In this paper, a fully coupled method is developed that includes a structural model to account for fluid structure interactions. The Euler or the Navier-Stokes equations and the structural equations are solved simultaneously in each real-time step. The elastic behavior of a blade is modeled with a linear spring for the bending motion and a torsional spring for the rotational degree of freedom. Quantitative comparisons are made between the results by the uncoupled and the coupled methods. A nonlinear effect, similar to that discovered by Carstens and Belz is studied in detail. Solutions by the Euler and Navier-Stokes calculations are also compared. The validity of the uncoupled method versus that of the coupled method is discussed to emphasize the potential dangers of using the uncoupled method for certain problems. 2. Parallel multi-passage Navier-Stokes solver A quasi-three-dimensional finite-volume method is used to calculate the flow through a cascade. For a two-dimensional

3 Computation of cascade flutter 561 control volume V with moving boundary V the Favreaveraged Navier-Stokes equations can be written as follows: ðð þ uðxþw dv þ uðxþðf ds x þ g ds y Þ t V V þ ðð ¼ uðxþðf m ds x þ g m ds y Þþ S dv ð1þ V where the vector w contains the conservative flow variables. The vectors f, f m, g, g m are the inviscid and viscous fluxes in the x- and y-directions, respectively. The method is quasithree-dimensional in the sense that the axial variation of the streamtube thickness u(x) is taken into account. The source term S is due to the variation of u in axial direction. A second-order accurate finite-volume scheme is used for spatial discretization. Equation (1) can then be written in semi-discrete form: dw þ RðwÞ ¼ dt ð2þ where R is the vector of residuals, consisting of the spatially discretized flux balance of equation (1). Time accuracy is achieved by using a second-order implicit time-discretization scheme which is recast into a pseudotime formulation, as proposed by Jameson (1991). The equations are marched in pseudo-time by a five-stage Runge-Kutta scheme, applying multigrid, local time stepping and residual smoothing. A multiblock grid with one block per blade-passage provides a natural domain decomposition which is utilized for parallel processing. The flow in each blade passage of the cascade is calculated on its own processor. The MPI is used to exchange boundary information between adjacent blade passages. Detailed descriptions of the method can be found in Ji and Liu (1999) and Sadeghi and Liu (21). 3. Structural model The structural behavior of a rigid blade profile with two degrees of freedom is governed by two coupled ODEs: m h þ S a a þ K h h ¼ 2L S a h þ I a a þ K a a ¼ M ea where h and a are the translational and rotational displacements, L and M ea are the aerodynamic force in direction of the displacement and the aerodynamic moment about the elastic axis, K h and K a are the bending and torsional spring stiffnesses, S a ¼ mbx a is the static unbalance of the profile, which provides the inertia coupling between the plunging and pitching degrees of freedom, I a ¼ mb 2 r 2 a is the area moment of inertia of the profile about the elastic axis, and m is the mass per unit span. The translational displacement h is non-dimensionalized with the semi-chord b. The pitching and plunging V ð3þ stiffnesses are modeled as stiffnesses of linear springs attached at the elastic axis. Time is non-dimensionalized with the eigenfrequency of the pitching mode, i.e. t ¼ v a t. The non-dimensional equations can be written in matrix form as where ½MŠ q þ½kšq ¼ F " # 1 x a ½MŠ ¼ ½KŠ ¼ x a r 2 a v h 6 v a r 2 a are the non-dimensional mass and stiffness matrices, and F ¼ 1 pmk 2 c ðv a=v h Þ 2 ( ) 2C l 2C m ; q ¼ ( ) h b a are the load and displacement vectors, C l and C m are the lift coefficient and the moment coefficient about the elastic axis, and k c ¼ v a c/2u 1 is the reduced frequency. The mass ratio m ¼ m/r 1 p b 2 can be regarded as a measure for the blade s inertial force compared to the aerodynamic forcing. The structural equation (4) are solved within the framework of a Rayleigh-Ritz modal approach, which is also applicable to general problems with a larger number of degrees of freedom. The structural motion is described by a linear combination of N eigenmodes: q ¼ XN r¼1 h r f r where f r is the r-th eigenvector of the generalized eigenproblem, and h r is the corresponding modal coordinate. If N is the total number of eigenmodes of the system, this description is an exact representation of the discrete problem. In cases with a large number of degrees of freedom, the problem size can be reduced by neglecting eigenmodes which do not significantly contribute to the response. In the present case of a twodimensional rigid body, we have only two eigenfrequencies corresponding to the two eigenmodes (symmetric and antisymmetric) which span the whole space. Therefore, no truncation is needed. The displacement vector can then be decomposed as q ¼ [f ] T {h}. Since the eigenvectors are orthogonal with respect to both the mass and stiffness matrices, premultiplying the governing equation (4) by [f ] T (normalized such that the eigenvectors are orthonormal with respect to the mass matrix) yields a set ð4þ

4 562 M. Sadeghi and F. Liu of equations in generalized coordinates of the form where h þ 2z i v i _h i þ v 2 i h i ¼ Q i ; i ¼ 1; 2;...; N ð5þ Q i ¼ f T i F; v2 i ¼ f T i ½KŠf i; f T i ½MŠf i ¼ 1 and z i is the modal damping of the i-th mode that has been added to the model. The assumption of a modal damping parameter is equivalent to introducing a classical damping matrix in equation (4) which is orthogonal to the eigenvectors. Following Alonso and Jameson (1994), the structural integrator is based on the decomposition of each of the modal equation (5) into a system of first-order differential equations. These equations are solved by a second-order implicit finite-difference method, applying a similar dualtime marching method as for the flow equations. Notice that the aeroelastic equations are implicitly coupled to the flow equations since Q i contains both C l and C m. Furthermore, the blade motion imposes unsteady boundary conditions on the flow equations. By simultaneously iterating the flow and structural equations within the pseudo-time approach, this coupling is taken into account. This coupling algorithm is also used for wing flutter calculations by Liu et al. (21). 4. Description of flutter simulation methods 4.1 Uncoupled method The term uncoupled points to the fact that the two-way coupling between structure and flow is not fully taken into account with this approach. The blades are externally forced to oscillate with a specified constant amplitude, frequency and interblade phase angle. Unsteady aerodynamic forces perform work on the blades. Positive work indicates that the aerodynamic moment amplifies the oscillation, in the case of negative work the moment stabilizes the motion. Stability is usually quantified by the amount of work performed within one cycle, i.e. the work coefficient or a related damping coefficient. The notion of positive or negative aerodynamic damping may seem to contradict the assumption of a strictly harmonic oscillation at constant amplitude. However, applying small amplitudes, the uncoupled approach constitutes a stability analysis in the sense that a particular motion is tested for stability under given flow conditions. At neutral stability, where accuracy is most important, the assumption of sinusoidal motion will indeed be adequate. In general, even far from the neutral point, the uncoupled approach is valid if the blade inertia, i.e. the mass ratio, is large. The frequency of the most susceptible structural eigenmode is usually assumed as the oscillation frequency. This assumption is valid at high mass ratios. 4.2 Coupled method In real systems, the blades are driven by the aerodynamic forces. Therefore, the motion of the blade will generally not follow a prescribed pattern with constant amplitude and interblade phase angle. The motion will depend on the mass ratio of the blade, its structural modes, as well as the flow conditions. The frequency of the oscillation will also be shifted from the structural eigenfrequency of the system, especially when the mass ratio is small. To account for this two-way coupling between structure and flow, the aeroelastic equations should be treated as a coupled system. Apart from flutter stability, a coupled method is also applicable to the study of limit-cycle oscillations, where the amplitude is limited by a nonlinear fluid structure interaction. In the following sections, we will contrast the range of validities of the uncoupled and coupled approaches. 5. Results and discussion An experimental test case by Buffum and Fleeter (199) is chosen for comparison between the uncoupled and coupled methods. A linear cascade of uncambered biconvex airfoils was tested for several interblade phase angles and two different reduced oscillation frequencies. The airfoils, with a thickness-to-chord ratio of 7.6% and with a stagger angle of 538, were oscillated in a pure rotational mode around mid-chord, with an amplitude of 1.28, an inlet Mach number of.65 and an incidence angle of 8. Buffum and Fleeter performed measurements with either all blades oscillating, or with only one oscillating blade and using the method of influence coefficients. They compared the experimental data with results from linearized theory for a flat plate. We first perform Euler calculations with the uncoupled method, i.e. the blades are forced to vibrate at prescribed torsional motions with given frequencies, amplitudes and interblade phase angles. 5.1 Uncoupled calculations Figures 1 and 2 show the magnitude and phase of the unsteady pressure difference between upper and lower surface for the reduced frequency of k ¼.223. Figure 3 shows the work coefficient calculated for various interblade phase angles (IBPA) at k ¼.223. A positive work coefficient indicates an unstable condition. The agreement of the Euler results with the data provided in Buffum and Fleeter (199) is reasonably good for both interblade phase angles of and 98. It appears that the Euler solutions are closer to those for the flat plate than to the experimental results, especially when comparing the work coefficient in figure 3.

5 Computation of cascade flutter 563 Magnitude of C P x/c Phase of C P Euler Exp. all blades oscillating * Flat plate * Exp. influence coeff. * * Buffum & Fleeter, x/c Figure 1. Unsteady pressure difference coefficient, IBPA ¼ 8, k ¼ Influence of fluid structure coupling on stability The influence of the mass ratio on the oscillation frequency and stability is investigated by solving the structural equations coupled with the aerodynamics. In the case of a single degree of freedom with pure torsion, the structural behavior is determined by specifying two parameters, here the torsional eigenfrequency v a and the mass ratio m of the blade: rffiffiffiffiffiffi K a v a ¼ I a m ¼ m pr 1 b 2 where K a and I a are the spring constant and moment of inertia, m is the blade mass per unit span, r 1 is the undisturbed density and b the half chord length. For a mass ratio of m ¼ 45, figure 4 shows the time history of the total energy of all blades in the cascade and the displacement angle of one of the blades at three different reduced eigenfrequencies, obtained with the coupled approach. In all three cases, the oscillation is initiated with the same total energy. One of the solutions is unstable with positive work increasing the total energy of the blades, one is neutrally stable with effectively no work done, and the highest frequency leads to a stable solution, where negative work decreases the blade energy. These calculations are performed with five blade passages, allowing for an interblade phase angle of 728 and its multiples, which is recognized as the most unstable IBPA in figure 3. The mass ratio determines how much the aerodynamic forcing influences the blade motion. With a high mass ratio, the aerodynamic forces have little effect on the eigenfrequency and eigenmode of the system. In this case of a single degree of freedom, the oscillation is purely harmonic at the reduced eigenfrequency k and with a constant amplitude. Therefore, the coupled solution with high mass ratio is expected to be close to the solution of the uncoupled method. However, as the mass ratio decreases, the solution of the coupled method will be different from the uncoupled result, and the actual oscillation frequency will deviate from the eigenfrequency of the blade. Figure 5 shows the reduced eigenfrequency and oscillation frequency at which the blade motion is neutrally stable, as a function of the mass ratio. In order to find the neutral frequency for a given mass ratio, the eigenfrequency is chosen by trial and error until the total energy is found to be constant in time, as was done for m ¼ 45 in figure 4. The blade oscillation frequency is then recorded as the neutral oscillation frequency shown in figure 5. Notice that for this case with only one Magnitude of C P x/c Phase of C P Euler Exp. all blades oscillating * Flat plate * Exp. influence coeff. * * Buffum & Fleeter, x/c Figure 2. Unsteady pressure difference coefficient, IBPA ¼ 98, k ¼.223.

6 564 M. Sadeghi and F. Liu k =.223 UNSTABLE STABLE Work coefficient C W neutrally stable at k =.334 Euler Exp. all blades oscillating * Flat plate * Exp. influence coeff. * * Buffum & Fleeter, oscillation mode in one degree of freedom, the actual oscillation frequency at which the motion is neutral does not depend on the structural parameters, but is unique. It is not significant how the blade is caused to oscillate at that unique neutral frequency, it can be either artificially forced by the uncoupled method, or resulting from aerodynamic-structural interaction. There is only one solution for which the aerodynamic forces perform zero work on the blades within each cycle. Figure 5 verifies that the lower the mass ratio is, the larger is the difference between the eigenfrequency and the oscillation frequency. Only if the mass ratio is large enough, those two frequencies are similar. At low mass ratios, e.g. with a light blade, the assumption that the blades will oscillate at their eigenfrequencies would lead to significant errors. For this test case, the neutral frequency is lower than the eigenfrequency. A system with an eigenfrequency in the shaded region of figure 5 would be erroneously claimed to be stable by a decoupled method while the coupled method correctly predicts it to be unstable. With the biconvex profile used in this study, realistic mass ratios for Inter-blade phase angle Figure 3. Work coefficient, k ¼.223. a solid blade are between m Ti ¼ 25 for titanium and m St ¼ 45 for steel (based on standard atmospheric density). In this range, the difference between eigenfrequency and oscillation frequency may be up to 5%, as seen in figure Nonlinear flutter Carstens and Belz also developed a computer code for simulating flutter in the time-domain (Carstens and Belz 21). They discovered a nonlinear phenomenon in a twodimensional cascade with a torsional degree of freedom. The same compressor cascade of NACA356 profiles, is used here for a detailed study of nonlinear cascade flutter. The profiles are staggered at an angle of 48 and the pitchto-chord ratio is.71. With an inlet Mach number of.9, an exit Mach number of.59 and an inlet flow angle of 48.38, the flow through the blade passages is transonic. Each profile is allowed to perform a solid body rotation about (x a /c ¼.5, y a /c ¼.2). While the eigenfrequency is fixed at 185 Hz, the blade mass ratio is chosen as a free Total Energy of the Cascade k =.281, unstable k =.34, neutrally stable k =.57, stable Deflection Angle.8º.4º.4º k =.281 k =.34 k = Time/Eigenperiod t/t Eigen.8º Time/Eigenperiod t/t Eigen Figure 4. Total energy of the cascade and angular displacements over time at m ¼ 45, for three different eigenfrequencies.

7 Computation of cascade flutter % 8º Mass Ratio = 16 Reduced Frequency at Neutral Stability Stable based on uncoupled theory, unstable based on coupled theory Eigenfrequency Stable 5% 4% 3% 2% 1%.35 % Oscillation Frequency Unstable.3 1% Mass Ratio Figure 5. Frequency at neutral stability over mass ratio. parameter to study the effects of nonlinear fluid structure interaction. Results for inviscid flow are obtained by solving the Euler equations. For viscous flow, the Navier-Stokes equations are solved applying the algebraic turbulence model by Baldwin and Lomax (1978). A simultaneous solution of the structural dynamics is obtained by the coupled approach. Figure 6 shows the steady-state results, i.e. results without blade motion, for inviscid and viscous flows. The pressure coefficient C p is plotted over the blade surface. Euler calculations are performed on two different grid resolutions. The result on the coarse grid agrees well with the data obtained on the fine grid. There are significant discrepancies between the viscous and the inviscid pressure distributions, especially concerning the location of the shock on the suction side. Since the blade profile is rather flat and slender, and the shock location is sensitive to the shape of the blade passage, even a small displacement thickness may have significant influence..6 4º 4º 8º 1 2 The dependence of the flutter behavior on the structural dynamics is investigated by obtaining solutions at varying blade mass ratio. Calculations are performed on four blade passages, therefore allowing inter-blade phase angles of, 9, 18 and Inviscid flow Figure 7. Angular displacement at mass ratio m ¼ 16. Figures 7 15 show the history of the displacement angles of the two adjacent blades for various mass ratios. These results are obtained by Euler calculations on four blade passages. The figures only show the oscillation for one pair of adjacent blades. The other two blades behave in a similar way. It is obvious from figure 7 that at a mass ratio of m ¼ 16, the inviscid cascade is unstable. Significant interaction between flow and structure dynamics at this low mass ratio causes the oscillation frequency in figure 7 to be about half the structural eigenfrequency. At a mass ratio of m ¼ 17, the cascade is still unstable but an interesting phenomenon occurs. In a time range of about 8 15 eigenperiods in figure 8, the odd numbered blades and the even numbered blades do not oscillate about the same mean displacement angle º Mass Ratio = 17 C P.2.4 Euler, 96 4 cells Euler, cells Navier-Stokes, cells 4º 4º x/c Figure 6. Steady state pressure coefficient over the blade Euler and Navier-Stokes results. 8º 1 2 Figure 8. Angular displacement at mass ratio m ¼ 17.

8 566 M. Sadeghi and F. Liu Mass Ratio = Time t / TEigen 4 5 Figure 9. Angular displacement at mass ratio m ¼ 181. The difference between the motions of neighboring blades becomes clearer at a higher mass ratio m ¼ 181 in figure 9, where curves of running average are shown to emphasize the short-time average displacement. These curves were obtained by filtering out the high flutter frequency. The blades temporarily oscillate around different average displacement angles (e.g. from t/teigen < 25 35) but then resume to oscillate about a common average (e.g. from t/teigen < 35 45). These two situations alternate at a frequency v1 that is significantly lower than the flutter frequency v2. Both frequencies are apparent in figure 9. Figure 1 shows Mach number contours for three different instances in time, marked by vertical dotted lines in figure 9. At an early stage (t/teigen < 6) there is a shock wave spanning the complete cross section of the lower blade passage, initiating the mode with different displacement angles. At t/teigen < 9.4, the flow is choked in every second blade passage, whereas in the other passages the two supersonic regions at the suction and pressure sides remain separate. After some time Mi =.9 Me =.59 µ = 181 t/teigen = 6. (e.g. t/teigen < 2.6), the blade motion returns to the original situation with a common average displacement angle and amplitude and an oscillation following a traveling wave with constant IBPA. Interestingly, this behavior exhibits neutral flutter stability and it remains neutral when the mass ratio is further increased. Figure 11, in comparison to figure 9, shows that the low frequency v1, at which the short-time average displacement angle is changing, is increasing with m. This also becomes clear from the Fourier transforms of the history of the displacement angles, shown in figure 12 for mass ratios between 181 and 19. The two dominant frequencies are the low frequency v1 and the high frequency v2, both of which are apparent in figures 9 and 11. Both frequencies are increasing with the mass ratio, thus approaching the blade eigenfrequency. In the limit of an infinite mass ratio one would expect the blades to oscillate with veigen. As the low frequency v1 increases, the corresponding amplitude decreases. Again, this is obvious from the amplitude of the running averages in figures 9 and 11. At mass ratios above 182, the oscillation mode with different average displacement angles dominates and at m ¼ 2, in figure 13, we identify a limit-cycle behavior leading to constant amplitude oscillation at two different mean displacement angles. A further increase of the mass ratio increases the time needed to reach a constant amplitude and also decreases the difference between the mean displacement angles. The result shown in figure 14 almost look steady and the cascade appears to reach stability in figure 15. However, even at relatively large mass ratios, there may be a finite time t* after which the nonlinear effect sets in so that neighboring blades will drift to new mean displacement levels. By separating the displacement levels of neighboring blades and finally causing oscillation, this nonlinear effect destabilizes the cascade. Consequently, the uncoupled approach may not apply in this case, even shaded regions are supersonic Figure 1. Mach number contours for m ¼ 181 at three time instances marked in figure 9.

9 Computation of cascade flutter 567 Mass Ratio = 185 Mass Ratio = Figure 11. Angular displacement at mass ratio m ¼ 185. Figure 13. Angular displacement at mass ratio m ¼ 2. though the mass ratio is relatively high, because the instability does not appear in terms of a linear flutter in traveling wave mode, but is a result of nonlinear fluid structure interaction. 5.5 Viscous flow Everything that was said about the Euler calculations also holds for the viscous solution shown in figures However, with viscous flow, the nonlinear effect appears at much lower mass ratios and also in a smaller range of mass ratios than in the case of inviscid flow. In other words, at comparable mass ratio, the viscous solution is more stable than the inviscid one. Figure 2 compares the average displacement of neighboring blades, in the range of mass ratios in which the nonlinear effect appears. Viscous and inviscid data agree in a qualitative sense. Starting from the lowest mass ratio at which the mean displacement angles of neighboring blades start to separate, the difference in the mean displacement angles increases as m is increased. At a certain mass ratio (m ¼ in the viscous case, m ¼ 22 in the inviscid case) the difference decreases as m is further increased. Mass Ratio = Figure 14. Angular displacement at mass ratio m ¼ 26. It was mentioned before that there are two frequencies involved, a high oscillation frequency v 2 and a lower beating frequency v 1. Figure 21 shows how these frequencies depend on the mass ratio. The higher frequency v 2 steadily increases, whereas v 1 exhibits a local maximum at about m ¼ 142 in the viscous case, and m ¼ 19 in the inviscid case. When m is beyond the local Deflection Amplitude ω 1 ω2 µ = 181 µ = 182 µ = 185 µ = 19 Mass Ratio = Oscillation Frequency ω/ω Eigen Figure 12. Frequency spectrum of displacement at various m Figure 15. Angular displacement at mass ratio m ¼ 4.

10 568 M. Sadeghi and F. Liu 3º Mass Ratio = Mass Ratio = 144 2º 2º 3º maximum, the beating amplitude decreases as m is increased. At higher mass ratios the beating vanishes entirely and the cascade exhibits a simple limit-cycle oscillation like those in figures 13 and 18 at the frequency v 2. Both frequencies are significantly lower than the blade eigenfrequency. However, as the mass ratio goes to infinity, v 2 is expected to approach the eigenfrequency. 6. Conclusions Figure 16. Angular displacement at m ¼ A coupled aerodynamics and structural dynamics method for cascade flutter simulation in the time domain is presented. Computations with the coupled method reveal significant differences between the natural frequencies of the blades and the actual flutter frequency at low mass ratios. It is demonstrated the possibility of false prediction of stability by the conventional uncoupled (energy) method because it determines a stability boundary by assuming that the blades oscillate at their natural frequencies. Furthermore, computations of a compressor cascade by the coupled method, solving either the Euler or the Navier-Stokes equations, show significant nonlinear effects for a range of mass ratios. At relatively low mass ratios, the coupled solutions show that the blades may Mass Ratio = Figure 18. Angular displacement at m ¼ 144. Mass Ratio = Figure 19. Angular displacement at m ¼ 15. oscillate alternatingly around the nominal design blade angle or a blade angle that is displaced from the design blade angle. The latter corresponds to a flow pattern with alternating choked and unchoked flow in neighboring Mean in º Mass Ratio, Navier-Stokes Solution Euler Navier-Stokes Blade 2 Blade Mass Ratio, Euler Solution Figure 17. Angular displacement at m ¼ 138. Figure 2. Time average displacement of adjacent blades.

11 Computation of cascade flutter 569 Oscillation Frequency ω/ω Eigen Mass Ratio, Navier-Stokes Solution Figure 21. blade passages. When this happens, the blades appear to oscillate with a major high frequency and a lower beating frequency, both lower than the structural eigenfrequency. When the mass ratio is increased, the beating amplitude decreases, and the separation of the mean pitch angles of neighboring blades becomes smaller until it vanishes for very large mass ratios. Such nonlinear phenomena cannot be predicted by the conventional uncoupled computations. Acknowledgements This work is supported by the Air Force Research Laboratory through the GUIde Program managed by the Carnegie Mellon University. References ω 1 inviscid ω 2 inviscid ω 1 viscous ω 2 viscous Mass Ratio, Euler Solution Low and high frequency components of nonlinear flutter. Abdel-Rahim, A., Sisto, F. and Thangam, S., Computational study of stall flutter in cascaded airfoils. J. Turbomach., 1993, 115, Alonso, J.J. and Jameson, A., Fully-implicit time-marching aeroelastic solutions. AIAA Paper 94-56, 1994 Bakhle, M.A., Reddy, T.S.R. and Keith, Jr., T.G., Time domain flutter analysis of cascades using a full-potential solver. AIAA J., 1992, 3, Baldwin, B.S. and Lomax, H., Thin-layer approximation and algebraic model for separated turbulent flows. AIAA Paper , 1978 Buffum, D.H. and Fleeter, S., Oscillating cascade aerodynamics by an experimental influence coefficient technique. J. Propulsion, 199, 6, Carstens, V. and Belz, J., Numerical investigation of nonlinear fluid structure interaction in vibrating compressor blades. J. Turbomach., 21, 123, Carta, F.O., Coupled blade-disc-shroud flutter instabilities in turbojet engine rotors. J. Eng. Power, 1967, 89, Clark, W.S. and Hall, K.C., A time-linearized Navier-Stokes analysis of stall flutter. J. Turbomach., 2, 122, Epureanu, B.I., Dowell, E.H. and Hall, K.C., Reduced order models of unsteady transonic flows in turbomachinery. J. Fluid. Struct., 2, 14, Hall, K.C. and Clark, W.S., A linearized euler analysis of unsteady transonic flows in turbomachinery. J. Turbomach., 1993a, 116, Hall, K.C. and Clark, W.S., Linearized Euler predictions of unsteady aerodynamic loads in cascades. AIAA J., 1993b, 31, Hall, K.C., Florea, R. and Lanzkron, P.J., A reduced order model of unsteady flows in turbomachinery. J. Turbomach., 1995, 117. Huff, D.L., Swafford, T.W. and Reddy, T.S.R., Euler flow predictions for an oscillating cascade using a high-resolution wave-split scheme. ASME Paper 91-GT-198, 1991 Hwang, C.J. and Fang, J.M., Flutter analysis of cascades using an Euler/Navier-Stokes solution-adaptive approach. J. Propul. Power, 1999, 15, Jameson, A., Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA Paper , 1991 Ji, S. and Liu, F., Flutter computation of turbomachinery cascades using a parallel unsteady Navier-Stokes code. AIAA J., 1999, 37(3), pp Lane, F., System mode shapes in the flutter of compressor blade rows. J. Aeronaut. Sci., 1956, 23, Liu, F., Cai, J., Zhu, Y., Tsai, H.M. and Wong, A.S.F., Calculation of wing flutter by a coupled fluid structure method. J. Aircraft, 21, 38(2), Moffatt, S. and He, L., Blade forced response predictions for industrial gas turbines, part I: methodologies. ASME Paper GT , 23 Sadeghi, M. and Liu, F., Computation of mistuning effects on cascade flutter. AIAA J., 21, 39(1), pp Sadeghi, M. and Liu, F., Investigation of mistuning effects on cascade flutter using a coupled method. AIAA Paper , 22 Sisto, F., Thangam, S. and Abdel-Rahim, A., Computational prediction of stall flutter in cascaded airfoils. AIAA J., 1991, 29,