39th AIAA Aerospace Sciences Meeting and Exhibit January 8 11, 2001/Reno, NV

Transcription

1 AIAA Investigatin f Nn-Linear Flutter by a Cupled Aerdynamics and Structural Dynamics Methd Mani Sadeghi and Feng Liu Department f Mechanical and Aerspace Engineering University f Califrnia, Irvine, CA th AIAA Aerspace Sciences Meeting and Exhibit January 8 11, 21/Ren, NV Fr permissin t cpy r republish, cntact the American Institute f Aernautics and Astrnautics 181 Alexander Bell Drive, Suite 5, Restn, VA

2 Investigatin f Nn-Linear Flutter by a Cupled Aerdynamics and Structural Dynamics Methd Mani Sadeghi and Feng Liu Department f Mechanical and Aerspace Engineering University f Califrnia, Irvine, CA A cmputatinal methd fr flutter predictin f turbmachinery cascades is presented. The flw thrugh multiple blade passages is calculated using a time-dmain apprach with cupled aerdynamic and structural mdels. The unsteady Euler/Navier- Stkes equatins are slved in quasi-3d using a secnd-rder implicit scheme with dual time-stepping and a multigrid methd. A structural mdel fr the blades with bending and trsin degrees f freedm is integrated in time tgether with the flw field. Infrmatin between structural and aerdynamic mdels is exchanged until cnvergence in each real-time step. The cmputatins are perfrmed n parallel cmputers, with each CPU perfrming the cmputatin f ne blade passage. Results fr a cmpressr cascade perating under transnic cnditins are presented. It is shwn that this test case exhibits strng nn-linear phenmena. In a certain range f blade mass ratis, the flw in each secnd blade passage is chked. In that situatin, neighbring blades scillate abut different average deflectins. The phenmenn is fund in inviscid flw as well as in viscus flw. Intrductin In cnventinal flutter calculatins the aerdynamic behavir and the structural behavir are treated in an uncupled way. Usually, a cascade is assumed t be tuned. All blades scillate with the same frequency and a cnstant inter-blade phase angle (IBP A). The flw arund blades scillating in this specified way is slved, and the Energy Methd is used t determine if the aerdynamic frces damp r amplify the blade mtin. In this apprach, frequency, amplitude and IBP A are fixed parameters that are nt affected by interactin between the fluid flw and the structural behavir. Hwever, aerelastic interactin can have imprtant influence and cause nn-linear phenmena that cannt be accunted fr with an uncupled methd. In additin, due t a fixed IBP A in the uncupled apprach, many calculatins have t be perfrmed in rder t investigate the pssibility f flutter at a given frequency and amplitude. On the ther hand, a single calculatin with a cupled apprach may simulate cnditins with a number f different IBP As, nly restricted by the number f blades used in the calculatin. The cmputatin will naturally pick up the mst unstable slutin. Sist et al. 1, 2 applied a cupled methd t study stall flutter in a linear cascade. These authrs used a Graduate Researcher, AIAA Student Member Assciate Prfessr, AIAA Member Cpyright c 21 by the authrs. Published by the American Institute f Aernautics and Astrnautics, Inc., with permissin. vrtex and bundary layer methd fr incmpressible flw cupled with a spring mdel fr the blade mtin. A similar trsinal spring and linear spring mdel fr rigid prfiles was used by Bakhle et al., 3 by Hwang and Fang, 4 and by Alns and Jamesn. 5 Bakhle et al. investigated ptential flw thrugh a cascade. A case with linear flw behavir was chsen in rder t validate the cupled methd by cmparisn with uncupled results. Hwang and Fang slved the Euler and Navier-Stkes equatins thrugh a cascade n unstructured grids. Alns and Jamesn develped a parallel methd fr flutter simulatin f islated wings. In a recent wrk, Carstens and Belz 6 discvered significant nn-linear effects with a transnic cmpressr cascade. They perfrmed numerical investigatins by slving the Euler equatins in the time dmain cupled with a structural mdel. The blades were scillating in a trsinal mde. At transnic flw cnditins, the cascade was fund t develp a flw pattern with chked flw and an scillating shck in each dd numbered passage, and purely subsnic flw in each even numbered passage. Ji and Liu 7 develped a Navier-Stkes cde t calculate quasi-three-dimensinal unsteady flws arund multiple scillating turbine blades. A time accurate multigrid methd is used applying the pseud-time apprach by Jamesn. 8 The cde was made parallel by using the Message Passing Interface MPI s that multiple passages can be calculated withut the use f phase-shifted peridic bundary cnditins fr blade flutter prblems. Therefre, the blade mtin des nt have t be peridic, neither spatially nr temp- f 12 American Institute f Aernautics and Astrnautics Paper

3 rally. Taking advantage f this prperty, the present authrs used this cde t study flutter f blades mistuned either in frequency r phase in an uncupled apprach. 9 In rder t perfrm fully cupled cmputatins, the present authrs further included a structural mdel t accunt fr fluid-structure interactins. 1 An integrated methd is used, which slves the Navier-Stkes and structural equatins simultaneusly in each realtime step. The elastic behavir f a blade is mdeled with a linear spring fr the bending mtin and a trsinal spring fr the rtatinal degree f freedm. A cmparisn between the uncupled and the cupled methd was made. 1 A nn-linear effect, similar t that discvered by Carstens and Belz, was shwn t appear als in an unstaggered NACA6 cascade with cupled bending and trsin in Euler calculatins. The present paper fcuses n a mre detailed study f this type f nn-linearity in transnic flw. Cmputatins are als extended t include the Navier- Stkes equatins and are cmpared with slutins by the Euler equatins. Parallel Multi-Passage Navier-Stkes Slver A quasi-three-dimensinal finite vlume methd is used t calculate the flw thrugh a cascade. Fr a tw-dimensinal cntrl Vlume V with mving bundary V the Favre-averaged Navier-Stkes equatins can be written as fllws: θ(x) w dv + θ(x) (f ds x + g ds y ) t V V = θ(x) (f µ ds x + g µ ds y ) + S dv V V (1) where the vectr w cntains the cnservative flw variables. The vectrs f, f µ, g, g µ are the Euler fluxes and viscus fluxes in the x- and y-directins, respectively. The methd is quasi-three-dimensinal in the sense that the streamtube thickness rati θ(x) is intrduced t accunt fr a variatin f the streamtube thickness in flw directin. The surce term S is due t the variatin f θ in x-directin. A secnd rder accurate finite-vlume scheme is used fr spatial discretizatin. Equatin (1) can then be written in semi-discrete frm: dw + R(w) = (2) dt where R is the vectr f residuals, cnsisting f the spatially discretized flux balance f Equatin (1). Time accuracy is achieved by using a secnd rder implicit time-discretizatin scheme which is recast int a pseud-time frmulatin, as prpsed by Jamesn: 8 dw dt + R (w) = (3) Fr efficiency the flw in each blade passage f the cascade is calculated n its wn prcessr. The Message Passing Interface (MPI) is used t exchange bundary infrmatin between adjacent blade passages. A detailed descriptin f the methd can be fund in References 7 and 9. Structural Mdel The structural behavir f a rigid blade prfile with tw degrees f freedm is gverned by tw cupled ODE s: mḧ + S α α + K h h = L S α ḧ + I α α + K α α = M ea (4) where h and α are the translatinal and rtatinal displacements, L and M ea are the aerdynamic frce cmpnent in directin f the displacement and the aerdynamic mment abut the elastic axis, K h and K α are the bending and trsinal spring stiffnesses, S α = mbx α is the static unbalance f the prfile, which prvides the cupling between the plunging and pitching degrees f freedm, I α = mb 2 r 2 α is the area mment f inertia f the prfile abut the elastic axis, and m is the mass per unit span. The translatinal displacement h is nn-dimensinalized with the semichrd b. The pitching and plunging stiffnesses are mdeled as stiffnesses f linear springs attached at the elastic axis. Time is nn-dimensinalized with the eigenfrequency f the pitching mde, i.e. τ = ω α t. In terms f these dimensinless quantities, the equatins f mtin fr the aerelastic systems are: d 2 dτ 2 ( h b x α d 2 dτ 2 ) d 2 α + x α ( ) h b dτ 2 + ω2 h h b = C l πµkc(ω 2 α /ω h ) 2 ω 2 α + rα 2 d 2 α dτ 2 + r2 αα = 2C m πµk 2 c(ω α /ω h ) 2 (5) where C l and C m are the lift cefficient and the mment cefficient abut the elastic axis, and k c = ω α c/2u is the reduced frequency. We rewrite the abve equatins in the mre familiar frm where [M] q + [K]q = F (6) [ ] 1 xα [M] = x α rα 2 [ ( ω h [K] = ω α ) 2 ] 2 r α are the nn-dimensinal mass and stiffness matrices, and { } { 1 h } Cl F = πµkc(ω 2 α /ω h ) 2, q = b 2C m α 2 f 12 American Institute f Aernautics and Astrnautics Paper

4 are the lad and displacement vectrs. In rder t slve Eq.(6), a Rayleigh-Ritz mdal apprach may be used. The mde shapes and frequencies are btained by slving the generalized eigenprblem assciated with the free vibratin prblem. In general, nly the first N mdes are cnsidered. With the first N mdes we have an apprximate descriptin f the displacement vectr f the system given by q = N η r φ r r=1 where φ r is the r-th eigenvectr f the generalized eigenprblem, and η r is the crrespnding nrmal crdinate. This truncated representatin yields a valid mdel if the mdal frequencies f the ignred mde shapes are much higher than the values f the aerdynamic frequencies invlved in the prblem. In the present case f a tw-dimensinal rigid bdy we have nly tw eigenfrequencies crrespnding t the tw eigenmdes (symmetric and antisymmetric) which span the whle space. Therefre, n truncatin is needed. The displacement vectr can then be decmpsed as q = [φ] T {η}. Since the eigenvectrs are rthgnal with respect t bth the mass and stiffness matrices, premultiplying the abve equatin by [φ] T (nrmalized such that the eigenvectrs are rthnrmal with respect t the mass matrix) yields a set f equatins in generalized crdinates f the frm where η + 2ζ i ω i η i + ω 2 i η i = Q i, i = 1, N (7) Q i = φ T i F, ω 2 i = φ T i [K]φ i, φ T i [M]φ i = 1 and ζ i is the mdal damping f the i-th mde that has been added t the mdel. The assumptin f the existence f a mdal damping parameter suppses that the damping matrix (which is nt included in Eq.6) is diagnalized by the apprpriate pre- and pstmultiplicatin by [φ] T and [φ]. Fllwing Alns and Jamesn, 5 The structural integratr is based n the decmpsitin f each f the mdal Equatins (Eq.( 7)) int a system f first-rder differential equatins. Using the transfrmatin x 1i = η i ẋ 1i = x 2i ẋ 2i = Q i 2ζ i ω i x 2i ω 2 i x 1i fr each f the mdal equatins, we can rewrite them in matrix frm as Ẋ i = [A i ]X i + F i, i = 1, 2 (8) where Ẋ i = { x1i x 2i } [, [A] = { F i = Q i 1 2ω i ζ i ω 2 i }. Nw, fr this system, assume anther transfrmatin X i = [P i ] 1 Z i, such that the new system ], Ż i = ([P i ] 1 [A i ][P i ])Z i + [P i ] 1 F i (9) is decupled, i.e. the matrices ([P i ] 1 [A i ][P i ]) are diagnal. Then fr each mde, the gverning equatins f mtin are dz 1i dτ = ω i( ζ i + dz 1i dτ = ω i( ζ i ζi 2 1)z 1i + ( ζ i + ζi 2 1) 2 ζi 2 1 Q i ζ 2 i 1)z 1i + (ζ i + ζ 2 i 1) 2 ζ 2 i 1 Q i (1) The time derivative peratr is nw discretized with a secnd rder scheme, the same scheme that is used t discretize the Navier-Stkes equatins, and the result is the fllwing set f tw difference equatins fr each mde 3z n+1 1i 3z n+1 2i 4z n 1i + zn 1 1i 2 t 4z n 2i + zn 1 2i 2 t = ω i ( ζ i + ζi 2 1)zn+1 1i + ( ζ i + ζi 2 1) 2 ζi 2 1 Q n+1 i = ω i ( ζ i ζi 2 1)zn+1 2i + (ζ i + ζi 2 1) 2 ζi 2 1 Q n+1 i (11) which can be integrated t steady-state in pseudtime, in the fllwing frm: where R 1i = R 2i = dz n+1 1i dt + R1i = dz n+1 2i dt + R2i = { } 3 2 t ω i( ζ i + ζ 2i 1) z n+1 1i ( ζ i + ζi 2 1) 2 ζi 2 1 Q n+1 i + S 1i { } 3 2 t ω i( ζ i ζ 2i 1) z n+1 2i (ζ i + ζi 2 1) 2 ζi 2 1 Q n+1 i + S 2i (12) 3 f 12 American Institute f Aernautics and Astrnautics Paper

5 and S ji = (z n 1 ji 4zji n )/(2 t) are surce terms frm the previus iteratin levels. Ntice that the aerelastic equatins are implicitly cupled t the flw equatins since Q i cntains bth C l and C m. By simultaneusly iterating the flw and structural equatins with the same pseud-time apprach, this cupling is taken int accunt. Fr a given Q n+1 i, the aerelastic equatins can be slved analytically fr the new values f z 1i and z 2i. Hwever, a five-stage Runge-Kutta scheme is used in rder t prvide fr the pssible future inclusin f nn-linearities in the structural mdel. Results and Discussin The same cmpressr cascade f NACA356 prfiles, that was studied by Carstens and Belz, 6 is used here t investigate nn-linear flutter. The prfiles are staggered at an angle f 4 and the pitch-t-chrd rati is.71. With an inlet Mach number f.9, an exit Mach number f.59 and an inlet flw angle f 48.3, the flw thrugh the blade passages is transnic. Each prfile is allwed t perfrm a slid bdy rtatin abut (x α /c =.5, y α /c =.2). While the eigenfrequency is fixed at 185Hz, the blade mass rati is chsen as free parameter t study the effects f nn-linear fluid-structure interactin. Results fr inviscid flw are btained by slving the Euler equatins. Fr viscus flw, the Navier-Stkes equatins are slved applying the algebraic turbulence mdel by Baldwin and Lmax. 11 A simultaneus slutin f the structure dynamics is btained by the methd described in the previus sectin. Figure 1 shws steady state results, i.e. results withut blade mtin, fr inviscid and viscus flws. The pressure cefficient C p is pltted ver the blade surface. Euler calculatins are perfrmed n tw different grid reslutins. The result n the carse grid agrees well with the data btained n the fine grid. There are significant discrepancies between the viscus and the inviscid pressure distributins, especially cncerning the lcatin f the shck n the suctin side. Since the blade prfile is rather flat and slender, and the shck lcatin is sensitive t the shape f the blade passage, even a small displacement thickness may have significant influence. The uncupled methd is used in a preliminary study as a basis fr cmparisn with cupled results. Figure 2 shws the damping cefficient Ξ ver the inter-blade phase angle fr the viscus and inviscid cases. A psitive damping cefficient signifies stability, whereas a negative damping cefficient indicates flutter. Using the uncupled apprach, ne wuld assume that the scillatin frequency equals the eigenfrequency f the blade. Hwever, if fluid-structure interactins are significant, the scillatin frequency will deviate frm the structural eigenfrequency. Therefre, a secnd result is included in Figure 2 (dtted lines), where the scillatin frequency was chsen t be half the eigenfrequency. When discussing the results f the cupled methd later in this wrk, it will becme clear that all investigated cases exhibit frequencies between.5 ω Eigen and ω Eigen. C P Euler, 96x4 cells Euler, 144x64 cells Navier-Stkes, 144x64 cells x/c Fig. 1 Steady state pressure cefficient ver the blade Euler and Navier-Stkes results. Damping Cefficient Ξ Euler, ω = ω Eigen Euler, ω =.5ω Eigen N.-S., ω = ω Eigen N.-S., ω =.5ω Eigen Inter-Blade Phase Angle Fig. 2 Damping cefficient ver IBP A by the uncupled methd fr viscus and inviscid flw at tw scillatin frequencies Accrding t the results f the uncupled apprach at an scillatin frequency f ω Eigen, i.e., disregarding fluid-structure interactins, the cascade is stable in the viscus case and almst neutrally stable in case f inviscid flw. At IBP A = 45 the inviscid cascade is slightly unstable with Ξ = The viscus and inviscid slutins fr the damping cefficient agree qualitatively but vary in magnitude. Fr the least stable IBP A f 45, the unsteady Euler result f the carse grid was fund t agree with calculatins n the fine grid. Therefre, the carse grid is chsen fr all fllwing inviscid calculatins. 4 f 12 American Institute f Aernautics and Astrnautics Paper

6 The dependence f the flutter behavir n the structural dynamics is investigated by btaining slutins at varying blade mass rati. The mass rati is defined as µ = m πρ b 2 (13) where m is the mass f the blade per unit span, ρ is the free stream density and b the half chrd f the blade. This nn-dimensinal parameter appears in the frcing term f the structural equatins and determines the influence f aerdynamic frcing n the blade mtin. The larger the mass rati, the mre dminant is the inertia f the blade, and the smaller is the influence by aerdynamic frces. The uncupled result in Figure 2 can be interpreted as the limit f the damping cefficient fr µ. At very large mass rati the influence f aerdynamic frcing n the structural behavir is negligible and the uncupled apprach is valid. A blade with a small mass rati, hwever, will react in strng accrdance t aerdynamic fluctuatins. Calculatins are perfrmed n fur blade passages, therefre allwing inter-blade phase angles f, 9, 18 and 27. At infinitely large mass rati, the cascade is stable at thse IBP As, as shwn in Figure 2. Hwever, when at lwer mass ratis fluidstructure interactins becme significant, the blades will nt scillate at their structural eigenfrequencies. Flutter instability at inter-blade phase angles between and 9 may be expected. Therefre, nw using the cupled apprach, we trigger the expected unstable mde by applying initial cnditins accrding t IBP A = 9. Cupled Results by Euler Calculatins Figures 3 thrugh 17 shw the deflectin histry f tw adjacent blades fr varius mass ratis. These results are btained by Euler calculatins n fur blade passages. The figures nly shw the scillatin fr ne pair f adjacent blades. The ther tw blades behave in a similar way. It is bvius frm Figure 3 that at a mass rati f µ = 16 the inviscid cascade is unstable. Qualitatively, this result is cnfirming the abve expectatin f destabilizing fluid-structure interactin at lw mass ratis. Hwever, it shuld be nted that the scillatin frequency in Figure 3 is smewhat higher than half the eigenfrequency. Fr that case, and fr an inter-blade phase angle f 9, the uncupled result in Figure 2 predicts flutter stability. Hence, the instability in Figure 3 cannt be explained by the uncupled methd, even when the actual scillatin frequency is knwn. Hwever, cmparisn between the uncupled and cupled methds is difficult in this case, because due t fluid-interactin the scillatin btained by the cupled methd is nt purely peridic, while the uncupled apprach impses a traveling wave mde with cnstant amplitude. Only at neutral stability, with cnstant amplitude and frequency, ne wuld expect bth results t agree. The results shwn in the fllwing figures, hwever, suggest that this type f neutral stability des nt exist with the investigated cascade. At a mass rati f µ = 17, the cascade is still unstable but an interesting phenmenn ccurs. In a time range f abut 8 t 15 eigenperids in Figure 4, the dd numbered blades and the even numbered blades d nt scillate abut the same mean deflectin. The difference between the mtins f neighbring blades becmes clearer at a higher mass rati µ = 18 in Figure 5, where curves f running average are shwn t emphasize the shrt-time average displacement. These curves were btained by filtering ut the high flutter frequency. Hwever, the blades still appear t return t the nrmal behavir f Figure 3, with an average displacement f and increasing amplitude. Increasing the mass rati just a little bit leads t the result shwn in Figure 6. The blades temprarily scillate arund different average deflectins (e.g. t/t Eigen 25 t 35) but then resume t scillate abut a cmmn average (e.g. t/t Eigen 35 t 45). These tw situatins are alternating at a frequency ω 1 that is significantly lwer than the flutter frequency ω 2. Bth frequencies are apparent in Figure 6. Figure 7 shws Mach number cnturs fr three different instances in time, marked by vertical dtted lines in Figure 6. At an early stage (t/t Eigen = 6) there is a shck wave spanning the cmplete crss sectin f the lwer blade passage, initiating the mde with different deflectin levels. In this situatin (e.g. t/t Eigen = 9.4 ), the flw is chked in every secnd blade passage, whereas in the ther passages the tw supersnic regins at the suctin and pressure sides remain separate. After sme time (e.g. t/t Eigen = 2.6), the blade mtin returns t the riginal situatin with a cmmn average deflectin and amplitude and an scillatin fllwing a traveling wave with cnstant IBP A. Interestingly, this behavir exhibits neutral flutter stability and it remains neutral when the mass rati is further increased. Figures 6 thrugh 1 shw that the lw frequency ω 1, at which the shrt-time average deflectin is changing, is increasing with µ. This als becmes clear frm the Furier transfrms f the deflectin histry, shwn in Figure 11 fr mass ratis between 181 and 19. The tw dminant frequencies are the lw frequency ω 1 and the high frequency ω 2, bth f which are apparent in Figures 6 thrugh 1. Bth frequencies are increasing with the mass rati, thus appraching the blade eigenfrequency. In the limit f an infinite mass rati ne wuld expect the blades t scillate with ω Eigen. 5 f 12 American Institute f Aernautics and Astrnautics Paper

7 As the lw frequency ω 1 increases, the crrespnding amplitude decreases. Again, this is bvius frm the amplitude f the running averages in Figures 6 thrugh 1. At mass ratis abve 182 the mde with different average deflectins dminates and at µ = 2, in Figure 12, we identify a limit cycle behavir leading t cnstant amplitude scillatin at different deflectin levels. A further increase f the mass rati increases the time needed t reach a cnstant amplitude and als decreases the difference f the deflectin levels (Figures 13 thrugh 17). The results shwn in Figures 15 and 16 almst lk steady and the cascade appears t reach flutter stability in Figure 17. The displacement histry in Figure 17 is what we wuld expect fr large mass ratis, knwing the uncupled result frm Figure 2. Hwever, even at relatively large mass ratis, e.g. Fig.16, there may be a finite time τ after which the nn-linear effect sets in s that neighbring blades will drift t new mean displacement levels. By separating the displacement levels f neighbring blades and finally causing scillatin, this nn-linear effect destabilizes the cascade. Cnsequently, the uncupled apprach wuld nt apply in this case, even thugh the mass rati is relatively high, because the instability des nt appear in terms f a linear flutter in traveling wave mde, but is a result f nn-linear fluid-structure interactin. - - Mass Rati = 16 Fig. 3 Angular displacement at mass rati µ = Mass Rati = 17 Fig. 4 Angular displacement at mass rati µ = Mass Rati = 18 Fig. 5 Angular displacement at mass rati µ = Mass Rati = 181 Fig. 6 Angular displacement at mass rati µ = f 12 American Institute f Aernautics and Astrnautics Paper

8 shaded regins are supersnic Mi =.9 Me =.59 µ = 181 t/teigen = 6. Fig Mach number cnturs fr µ = 181 at three time instances marked in Fig.6 Mass Rati = Mass Rati = Time t / TEigen Fig Time t / TEigen Angular displacement at mass rati µ = 182 Fig. 9 Angular displacement at mass rati µ = f 12 American Institute f Aernautics and Astrnautics Paper

9 Mass Rati = 19 Mass Rati = Fig. 1 Angular displacement at mass rati µ = 19 Fig. 13 Angular displacement at mass rati µ = 22 Deflectin Amplitude ω 1 ω2 µ = 181 µ = 182 µ = 185 µ = Oscillatin Frequency ω/ω Eigen Fig. 11 Frequency spectrum f displacement at varius µ - - Mass Rati = 24 Fig. 14 Angular displacement at mass rati µ = 24 Mass Rati = 2 Mass Rati = Fig. 12 Angular displacement at mass rati µ = 2 Fig. 15 Angular displacement at mass rati µ = 26 8 f 12 American Institute f Aernautics and Astrnautics Paper

10 - - Mass Rati = 3 Fig. 16 Angular displacement at mass rati µ = Mass Rati = 4 Fig. 17 Angular displacement at mass rati µ = 4 Cupled Results by Navier-Stkes Calculatins Everything that was said abut the Euler calculatins als hlds fr the viscus slutin shwn in Figures 18 thrugh 23. Hwever, with viscus flw, the nn-linear effect appears at much lwer mass ratis and als in a smaller range f mass ratis than in the case f inviscid flw. In ther wrds, at cmparable mass rati, the viscus slutin is mre stable then the inviscid ne. Frm the uncupled results in Figure 2 we wuld expect the Navier-Stkes slutin t be slightly less stable than the Euler slutin at an IBP A = 9. Hwever, a cmparisn between uncupled and cupled results is nly meaningful at a pint f neutral stability, where the blades scillate in a traveling wave mde. Due t the nn-linear behavir, this type f neutral stability des nt exist. The uncupled methd des nt accunt fr nn-linearity in time, s that it is nt surprising that in this case the tw different methds d nt exactly agree n the pint f neutral stability. Figure 24 cmpares the average displacement f neighbring blades, in the range f mass ratis in which the nn-linear effect appears. Viscus and inviscid data agree in a qualitative sense. Starting frm the lwest mass rati at which the displacement level f neighbring blades starts t separate, the difference in displacement level increases as µ is increased. At a certain mass rati (µ = in the viscus case, µ = 22 in the inviscid case) the difference decreases as µ is further increased. It was mentined befre that there are tw frequencies invlved, a high scillatin frequency ω 2 and a lwer beating frequency ω 1. Figure 25 illustrates hw these frequencies depend n the mass rati. The higher frequency ω 2 steadily increases, whereas ω 1 exhibits a lcal maximum at abut µ = 142 in the viscus case, and µ = 19 in the inviscid case. When µ is beynd the lcal maximum, the beating amplitude decreases as µ is increased. At higher mass ratis the beating entirely vanishes and the cascade exhibits a simple limit cycle flutter like in Figures 12 and 22 at the frequency ω 1. Bth frequencies are significantly lwer than the blade eigenfrequency. Hwever, as the mass rati ges t infinity, ω 2 is expected t apprach the eigenfrequency. 9 f 12 American Institute f Aernautics and Astrnautics Paper

11 3 Mass Rati = Mass Rati = Fig. 18 Angular displacement at µ = Fig. 21 Angular displacement at µ = 14 Mass Rati = 137 Mass Rati = Fig. 19 Angular displacement at µ = Fig. 22 Angular displacement at µ = 144 Mass Rati = 138 Mass Rati = Fig. 2 Angular displacement at µ = Fig. 23 Angular displacement at µ = 15 1f 12 American Institute f Aernautics and Astrnautics Paper

12 Mean in Mass Rati, Navier-Stkes Slutin Euler Navier-Stkes Blade 2 Blade Mass Rati, Euler Slutin Oscillatin Frequency ω/ω Eigen Mass Rati, Navier-Stkes Slutin ω 1 ω 2 ω 1 ω 2 inviscid inviscid viscus viscus Mass Rati, Euler Slutin Fig. 24 blades Time average displacement f adjacent Results with a Larger Number f Blades The uncupled result in Figure 2 implies that the mst unstable IBP A is clse t 45. Therefre, we perfrm calculatins n eight blade passages in rder t allw fr mre unstable cnditins than with the abve studies n fur blades. Figures 26 and 27 shw the angular displacements f neighbring blades frm Euler calculatins at µ = 185 and frm Navier- Stkes calculatins at µ = 138. These tw mass ratis were chsen fr the nn-linearity t be at a similar stage in the viscus and inviscid cases (cmpare the inviscid and viscus slutins n fur blade passages in Figures 9 and 2). Cmparisn f Figures 26 and 27 with the results n fur blade passages (Fig.9 and Fig.2) shws that the cascade is mre unstable at the larger number f blades. The nn-linear behavir is still apparent in the slutin n eight passages, but the average displacement diverges frm zer nly ver a small initial range f time. This is true fr bth the viscus and the inviscid case. Figures 28 and 29 shw the histry f angular displacement at higher mass ratis calculated n eight passages. Fr the mass ratis f µ = 3 in the inviscid case and µ = 15 in viscus flw, the slutin n fur blade passages is stable as seen in Figures 16 and 23. The inviscid slutin n eight passages (Fig.28), allwing fr inter-blade phase angles f 45 and its multiples, appears t be unstable. The viscus slutin in Figure 29, hwever, is still stable even n eight blades. In bth cases there is qualitative agreement with the uncupled result fr.5ω Eigen in Figure 2, since at lw frequency (due t fluid-structure interactin) the mst unstable IBP A is 45 in inviscid flw, but 9 in viscus flw. Therefre, the inviscid cascade may be expected t be less stable when allwing fr IBP A = 45, while the stability f the viscus Fig. 25 Lw and high frequency cmpnent f nn-linear flutter case is already at its minimum at IBP A = 9. Of curse, the results in Figure 2 d nt accunt fr fluidstructure interactin that lead t instabilities thrugh ther mechanisms than thrugh the change f scillatin frequency. Cnclusins A cupled aerdynamics and structural dynamics methd fr flutter simulatin is presented. A cmpressr cascade f the NACA356 airfil with ne degree f freedm in pitching arund mid-chrd in a transnic flw is investigated. Bth Euler and Navier-Stkes calculatins are perfrmed. Uncupled calculatins by the energy methd predict that the blades are stable fr IBP A = 9. Cmputatins by the cupled methd with fur blade passages, hwever, shw significant nn-linear effects fr a range f mass ratis f the blades. When the mass rati is small, the cascade is unstable. As the mass rati increases, the cupled slutins shw that the blades may scillate alternatingly arund the nminal design blade angle r a blade angle that is displaced frm the design blade angle. The latter crrespnds t a flw pattern with alternating chked and unchked flw in neighbring blade passages. When this happens, the blades appear t scillate with a majr high frequency and a lwer beating frequency, bth lwer than the structural eigenfrequency. Althugh the blades exhibit this cmplex nn-linear scillatry behavir, they are neutrally stable. When the mass rati is further increased, the beating amplitude decreases, and the separatin f the displacement f the mean pitch angles f neighbring blades becmes smaller until it vanishes eventually when the mass rati is very large. At that time, the slutin f the cupled apprach is in agreement with that by the decupled energy methd. 1f 12 American Institute f Aernautics and Astrnautics Paper

13 6 Mass Rati = 185, EULER.6 Mass Rati = 3, EULER Fig. 26 Angular displacement at mass ratis a) µ = 185 (inviscid) and b) µ = 138 (viscus), calculated n eight blade passages 6 Mass Rati = 138, NAVIER-STOKES Fig. 28 Angular displacement at mass ratis a) µ = 3 (inviscid) and b) µ = 15 (viscus), calculated n eight blade passages.6 Mass Rati = 15, NAVIER-STOKES Fig. 27 Angular displacement at mass ratis a) µ = 185 (inviscid) and b) µ = 138 (viscus), calculated n eight blade passages The nn-linear behavir is fund by bth the Euler and Navier-Stkes calculatins. Hwever, in case f the Navier-Stkes calculatins the nn-linearity appears at lwer mass ratis and in a smaller range f mass ratis than with Euler calculatins. Further investigatin f the characteristics, mechanisms, and significance f such phenmena is needed. References 1 Sist, F., Thangam, S., and Abdel-Rahim, A., Cmputatinal Predictin f Stall Flutter in Cascaded Airfils, AIAA Jurnal, Vl. 29, 1991, pp Abdel-Rahim, A., Sist, F., and Thangam, S., Cmputatinal Study f Stall Flutter in Cascaded Airfils, Jurnal f Turbmachinery, Vl. 115, 1993, pp Bakhle, M. A., Reddy, T. S. R., and Jr., T. G. K., Time Dmain Flutter Analysis f Cascades Using a Full-Ptential Slver, AIAA Jurnal, Vl. 3, 1992, pp Hwang, C. J. and Fang, J. M., Flutter Analysis f Cascades Using an Euler/Navier-Stkes Slutin-Adaptive Ap- Fig. 29 Angular displacement at mass ratis a) µ = 3 (inviscid) and b) µ = 15 (viscus), calculated n eight blade passages prach, Jurnal f Prpulsin and Pwer, Vl. 15, 1999, pp Alns, J. J. and Jamesn, A., Fully-Implicit Time- Marching Aerelastic Slutins, AIAA Paper 94 56, Carstens, V. and Belz, J., Numerical Investigatin f Nnlinear Fluid-Structure Interactin in Vibrating Cmpressr Blades, ASME Paper 2 GT 381, 2. 7 Ji, S. and Liu, F., Flutter Cmputatin f Turbmachinery Cascades Using a Parallel Unsteady Navier-Stkes Cde, AIAA, Vl. 37, 1999, pp Jamesn, A., Time Dependent Calculatins Using Multigrid, with Applicatins t Unsteady Flws Past Airfils and Wings, AIAA Paper , Sadeghi, M. and Liu, F., Cmputatin f Mistuning Effects n Cascade Flutter, AIAA Paper 2-23, 2, t appear in AIAA Jurnal. 1 Sadeghi, M. and Liu, F., Cmputatin f Cascade Flutter with a Cupled Aerdynamic and Structural Mdel, t be published in the Prceedings f the 9 th ISUAAAT, Lyn, France, Sept Baldwin, B. S. and Lmax, H., Thin-Layer Apprximatin and Algebraic Mdel fr Separated Turbulent Flws, AIAA Paper , f 12 American Institute f Aernautics and Astrnautics Paper