GT NUMERICAL STUDY OF A CASCADE UNSTEADY SEPARATION FLOW

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1 Proceedings of ASME urbo Exo Power for Land, Sea, and Air Vienna, Austria, June 4-7, 2004 G NUMERICAL SUDY OF A CASCADE UNSEADY SEPARAION FLOW Zongjun Hu and GeCheng Zha Deartment of Mechanical Engineering University of Miami Coral Gables, FL 3324 J. Leicovsky QSS Grou, Inc. Cleveland, Ohio 4435 ABSRAC A CFD solver is develoed to solve a 3D, unsteady, comressible Navier-Stokes equations with the Baldwin-Lomax turbulence model to study the unsteady searation flow in a high incidence cascade. he second order accuracy is obtained with the dual time steing technique. he code is first validated for its unsteady simulation caability by calculating a 2D transonic inlet diffuser flow. hen a 3D steady state calculation is carried out for the cascade at an incidence of 0. he surface ressure distributions comare reasonably well with the exeriment measurement. Finally, the 3D unsteady simulation is carried out with 3 inlet Mach numbers at the incidence of 0. he searation bubble oscillation and the static ressure oscillation on the leading edge of the blade suction surface exhibit clear eriodicity. he details of the leading edge vortex shedding is catured. he inlet Mach number is shown to be the determinant factor in the characteristics of the searation flow. In the subsonic inlet flow region, increasing the inlet Mach number enlarges the searation region and the ressure oscillation intensity. he searation flow is weakened when the inlet flow becomes suersonic. NOMENCLAURE ρ density µ µ t molecular, turbulent viscoscity τ stress γ secific heat ratio ω vorticity PhD Student, zongjunhu@yahoo.com Associate Professor, Director of CFD Lab Senior Research Suervisor a local seed of sound e energy er unit mass l mixing length static ressure u v w velocity comonents x y z Cartesian coordinates t time C static ressure coefficient E F G inviscid flux vector Ma Mach number Pr Prandtl number Q conservative variable vector Re Reynolds number R S viscid flux vector Introduction Flutter in axial turbomachines is a highly undesirable and dangerous self-excited blade oscillation mode that can result in high cycle fatigue blade failure. Modern turbine engines emloy transonic fan stages with high asect ratio blades that are rone to flutter. It is imerative to understand the origins of flutter for reliable and safe oeration of these engines. High subsonic and transonic torsional stall flutter occurs near the fan stall limit line at seeds u to about 80% of the design seed and with high incidence. wo otential factors are assumed to trigger the flutter mode, the shock wave motion in the transonic conditions and the large searation on the suction side of the blade surface under high incidence angle []. Actually, the shock wave does not aear until very high subsonic inlet Mach number is reached. However, the flutter exists in much Coyright c 2004 by ASME

2 wider Mach number region (aears in smaller Mach number) than where the shock wave exists. he searation flow on the leading edge of the blade suction side is a likely cause for flutter. A series of exeriments have been carried out in NASA Glenn Research Center to study the transonic searation flow characteristics of modern airfoils for transonic fans. A low asect ratio fan blade oerating near the stall flutter boundary under high incidence is simulated in the NASA ransonic Flutter Cascade. he unsteady ressure is measured at selected oints on the chord line of the cascade surfaces [2]. he flow attern is visualized using dye oils and schilieren flow visualization methods []. he objective of this aer is to numerically study the unsteady characteristics of the NASA transonic cascade searation flow with a incidence angle of 0 in 3D condition. A CFD solver is develoed to solve the 3D, unsteady, comressible Navier- Stokes equations with the Baldwin-Lomax turbulence model [3]. he dual time steing method is alied to achieve the second order accuracy in time. he unsteady comuting caability of the CFD solver is validated by a transonic inlet diffuser flow, where turbulent boundary layer interacts with the shock wave and causes unsteady flow searation. Finally, the unsteady searation flow simulation is carried out at 3 inlet Mach numbers, 0.5, 0.8 and.8. he vortex shedding mechanism is analyzed for the case of Mach number 0.5. he characteristics of the searation flow under varying inlet Mach numbers is demonstrated. Numerical Algorithm he governing equations for flow field simulation used in this aer are the Reynolds averaged time-deendent comressible Navier-Stokes equations in generalized coordinate system. he simulation is carried out with finite volume method. he equations are discretized using the third order MUSCL differencing [4]. he linearized equation systems are solved using the Gauss-Seidel line iteration. Uwind differencing is imlemented with the Roe scheme [5] and the van Leer scheme [6]. he second order accuracy of time marching is obtained with the dual time steing technique. Governing Equations For simlicity, the non-dimensional form of the equations in conservation law form are exressed in Cartesian coordinates as the followings. E ρũ ρũ 2 ρũṽ ρũ w ρẽ ũ F ρṽ ρũṽ ρṽ 2 ρṽ w ρẽ ṽ G ρ w ρũ w ρṽ w ρ w 2 ρẽ w R 0 τ xx τ xy τ xz β x Re S 0 τ xy τ yy τ yz β y Re 0 τ xz τ yz τ zz β z Re he shear stress terms are exressed as, 2 τ i j 3 µ ũ k µ t δ i j x k β x β y and β z are exressed as, β i µ µ t ũ j τ i j γ µ µ t ũ i x j ũ j x i µ µ t ã 2 Pr Pr t x i In above equations, ρ is the density, u v w are the Cartesian velocity comonents in x y z directions, is the static ressure, and e is the energy er unit mass, a is the local seed of sound. he overbar denotes a Reynolds average, and the tilde is used to denote the Favre mass average. he molecular viscosity µ is determined by the Sutherland law. he Reynolds stresses are related to mean flow variables through a turbulent viscosity µ t based on the Boussinesq assumtion, and the turbulent viscosity µ t is determined by the Baldwin-Lomax model [3]. where t E F G x y z R S x y z () Baldwin-Lomax urbulence Model [3] he turbulent viscosity µ t is comuted searately in two layers. For clarity, the overbar and tilde are omitted in the following. At inner layer, Q ρ ρũ ρṽ ρ w ρẽ µ ti ρl 2 ω (2) 2 Coyright c 2004 by ASME

3 where l ky ex y A (3) Discretization Method he governing equations () are discretized and solved using finite volume method. he equations are rewritten as, are the dimensional and dimen- ω is the local vorticity, y and y sionless distance to the wall. At the outer layer, t E R x Using finite volume method, F y S G z (7) µ to KC c ρf wake F kleb (4) dv R F s t s F wake min y max F max C wake y max u 2 diff F max where, dv is the volume of cell, s is cell interface area vector in the normal outward ointing direction, and, R F E R i F S j G k F kleb 5 5 C kleby y max 6 Discretize this equation in imlicit form with first order differencing in both time and sace. he discretized equations at cell (i j k) are written as the following, In the above formulations, k, A, C wake, C kleb, C c and K are constants. he quantities u diff, F max and y max are determined by the velocity rofile following a line normal to the wall. F max and y max are the maximum value and the corresonding distance of function F y, where, Q A Q i B Q j C Q k A Q A Q i B Q B Q j C Q C Q k R (8) F y y ω ex y A (5) R t R F ds dv s n u diff u 2 v 2 w 2 max u 2 v 2 w 2 min In the 3D comutation of this aer, the wall is located at the 4 side of the comutation domain. he value of µ t is simly comuted according to the closest wall surface. F max and y max is searched from the wall to the center of the assage. In the wake region, the exonential art is set to zero in Eq. (3) and Eq. (5). he second art of Eq. (6) is zero outside of the wake region. All constants are assigned the standard values suggested in [3]. he turbulence model is alied and validated in [7] for a subsonic turbulent flat late boundary layer flow. (6) Q Q n where, n, n denote two sequential time stes. Equation (8) is solved using the line Gauss-Seidel iteration method. he convective fluxes E, F, G are evaluated by the Roe scheme [5] or the van Leer scheme with MUSCL differencing. hird order differencing is used for convective terms E, F, G and second order central differencing is used for the viscous terms R, S,. Local time ste is alied to seed u the convergence. ime Marching he dual time steing technique is used to obtain the second order accuracy in time. he method is made ossible by Q n 3 Coyright c 2004 by ASME

4 adding a seudo temoral term τ to Eq. (). he governing equation becomes, E R x τ t F S y G z where τ is the seudo time and t is the real time. he real time ste t is secified by the hysical time interval. he solution iteration is oerated on the seudo time ste τ, which is limited by the CFL criterion. When the imlicit solution is converged on seudo time τ iteration, one hysical time ste t is finished. Numerical techniques, such as local time ste, Gauss-Seidel method, can still be used to seed u the convergence in the iteration on the seudo time ste τ. Write Eq. () in integral form, τ t (9) R (0) Alying the second order discretization for t, t 3Q n 4Q n 2 t Q n where n is the hysical time ste. he value at time ste n is unknown and will be solved by the seudo time iteration. At the unknown time ste n, using Euler discretization for the seudo temoral term, let m and m be consequential seudo time stes. τ Q n m Q n m τ he governing equations with seudo temoral term Eq. (0) are imlicitly written as, herefore, Q n m Q n m 3Q n 4Q n Q n τ 2 t R n m () 5 τ t I R R n m n m δq n m 3Q n m 4Q n Q n 2 t (2) Eq. (2) is iteratively solved using the Gauss-Seidel line iteration methods. Q n m is assumed to be equal to Q n at the beginning of the iterations. Results and Discussion In this section, the unsteady simulation of the code is first validate by a transonic inlet-diffuer flow. hen the numerical study is carried out for the casecade searated flow with an incidence angle of 0 under 3 inlet Mach numbers, 0.5, 0.8 and.8. he cascade exeriment measurement is reorted in [2, 8]. Unsteady ransonic Inlet-diffuser Simulation A 2D transonic inlet-diffuser internal flow is first calculated to validate the unsteady simulation ability of the current code. his case is exerimentally measured in [9] to study the frequency characteristics of shock wave oscillation resulting from the interaction between the turbulent boundary layer and the shock wave. he test section is designed to obtain a 2D flow condition. he flow enters the inlet diffuser under subsonic condition and accelerates to suersonic at the throat. By adjusting the back ressure level at outlet, different shock wave structures are obtained. Fig. shows the the mesh structure. he inlet diffuser has a height of h t 4.4 cm at the throat and a total length of 2.6h t. he inlet height is h in 4h t. he mesh size is 30(x) 60(y). he grid is uniformly distributed in horizontal direction before and after the throat section. In the throat region, the grid is clustered to catch the shock wave sharly. he mesh is also clustered close to the uer and bottom walls to make sure the maximum y is below 3. he Roe scheme is alied to evaluate the control volume inter surface flux. he total ressure t, total temerature t and flow angle α are secified at the inlet. he no-sli adiabatic boundary condition is alied at to and bottom walls. he back ressure level is set as outlet t =0.72 as the exeriment. he throat height is used as the characteristic length. he Reynolds number is he static ressure is fixed at the outlet. he hysical time ste is set as ms, which is about 7% of the shock oscillation cycle. he Gauss-Seidel solution is carried out between two sequential hysical time stes with a CFL of 5.0. he converged solution for each hysical time ste is obtained after 50 seudo time ste iterations. he obtained inlet Mach number is he instant Mach contours are shown in Fig. 2. he shock wave is clearly catured. he flow is searated after the shock, which brings high unsteadiness to the flowfield. he location of the shock wave moves back and forth downstream of the throat region. he flow field arameters, including the shock wave location and static ressure, vary eriodically with time in the region after the shock wave. Fig. 3 shows the ressure oscillation 4 Coyright c 2004 by ASME

5 history on the uer wall at outlet. More clear frequency information is revealed using the Fast Fourier ransform (FF) technique. he static ressure frequency sectrum is shown in Fig. 4 on the left comared with exeriment measurement based on the shock wave motion [9] on the right. he comuted frequency sectrum eak is at 250 Hz, which is close to the exeriment measurement frequency, 200 Hz. he reason for the discreancy between the comutational and exerimental results may be the inadequacy of the turbulence model, which will affect the boundary layer thickness and hence the shock motion. 5 mesh oints allocated before, on and after the blade surface. Fig. 9 shows the the mesh structure and the geometry of the blade at bottom, mid-san and to lanes. he mesh is clustered in regions close to the blade surface on the η direction and in regions close to the to and the bottom end walls in the ζ direction. In ξ direction, the mesh is clustered in regions close to the leading and trailing edges of the blade, where the flow structure is comlicated. he maximum y at all wall boundaries is under 3. For clarity, every one of two grids lines are lotted in Fig. 8 and Fig. 9. 3D Cascade Geometry, Meshing and Boundary Conditions he test section of the NASA transonic flutter cascade facility is shown in Fig. 5. Nine blades were located in the test section. With a reference to the axial direction, the setting angle of the inlet duct is 20, and the blade-setting angle is 30. his results in 0 chordal incidence for the airfoils. Fig. 6 shows the mid-san cross section of the cascade. he test section has a rectangular cross section of 5.84 cm wide (itch s) by 9.59 cm high (height h). he aerodynamic chord c is 8.89 cm with a maximum thickness of 0.048c at 0.625c from the leading edge. his results in a solidity c s of.52. he exeriment blades have constant cross section in san-wise direction, excet near the end walls, where they have large, diamond-shaed fillets to suort the attachment shafts. he blades are not exactly symmetric about the mid-san lane. he fillet on the drive-side is larger than the one on the free side (Fig. 9). his makes the 3D calculation necessary. he current simulation is carried for the itch-wise central assage of the cascade. he cascade shows good eriodicity in flowfield measurement [0]. he mid-san comutational domain is shown in Fig. 7. In the lot, to make x axis ointing to right horizontally, the blade is rotated 90 counter-clockwise relative to Fig. 5 and 6. his is followed by the rest figures in this aer. o minimize the influence of boundary condition secification, the comutation domain is stretched outside for.2 chord length in stream-wise direction at inlet and outlet. Boundaries, A B, C D, E F and G H are set as eriodic boundaries. No-sli adiabatic boundary condition is alied at the wall surface B C and F G. otal ressure P t, total temerature t and flow angles α, β are given at inlet boundary A H uniformly. At the outlet boundary D E, a constant static ressure is fixed. he no-sli adiabatic boundary condition is also alied on the two end walls at bottom and to. he general coordinates is used in the simulation with ξ axis aligning with the stream-wise direction, η axis aligning with the itch-wise direction and ζ axis following the san-wise direction. he three dimensional mesh structure is shown in Fig. 8. he end walls are located at the bottom and the to. he mesh size is 00(ξ) 60(η) 60(ζ). In ξ direction, there are 5, 70 and Steady state results hough the flow is searated and unsteady when the incidence is high, the time averaged flow field is calculated with the local time steing enabled and the dual time steing disabled in the code. he solution is obtained when the calculated flowfield is unchanged. he steady state solution is also used as the initial solution for the corresonding unsteady calculation. An incidence of 0 is chosen for the following numerical study. he van Leer scheme is used to evaluate the inviscid flux. hough the van Leer scheme is more diffusive than the Roe scheme, when working with the Baldwin-Lomax turbulence model, it gives better agreement with the exeriment in the current code. he Roe scheme redicts the searation larger than the exeriment. he inlet Mach number is obtained by adjusting the back ressure level. he result of the case with inlet Ma=0.5 is described in this section to demonstrated the characteristics of the 3D searation flow field. More results for Ma=0.8 and Ma=.8 can be found in [7]. In the case of Ma=0.5, the corresonding Reynolds number is he CFL in the Gauss-Seidel iteration is 5.0. he calculation starts from a flowfield at rest. he flow stream lines on the mid-san lane is shown in Fig. 0. he flow exhibits a large searated region on the suction surface that starts immediately at the leading edge and extends down to 45% of the blade chord. he flow attern on the suction surface is lotted on the left in Fig.. he searation region has a arabola shae, which is aroximately symmetric about along the blade mid-san line and extends to the blade ustream corners. wo counter rotating vortexes are formed downstream of the blade leading edge corners on the suction surface at its two ends. he exeriment visualization with dye oil technique is shown in Fig. on the right. he comutation results agree with the exeriment fairly well, excet that the numerical results shows a fuller searation region in san-wise direction. he mid-san static ressure distribution is lotted and comared with the exeriment measurement in Fig. 2. A reasonable agreement is achieved. he ressure is exressed as the ressure 5 Coyright c 2004 by ASME

6 coefficient, C in 2 ρ inuin 2 where is the local static ressure. in ρ in and U in are the averaged static ressure, density and velocity at the inlet. On the leading edge of the suction side, the numerical ressure results varies more steely than that given by the measurement. he searation region denoted by the cross of the ressure distributions on suction and ressure surfaces agrees very well with the exeriment. he searation region is enlarged when Roe scheme is used to calculate the inviscid flux. A ossible reason for the difference is the alication of turbulence model on the H-tye mesh in the current code. It is shown that the imlementation details the Baldwin-Lomax model is vital to the resulted turbulent viscosity accuracy. Better agreement is obtained in [] for the same case, where the Roe scheme is alied on an O-tye mesh. he H-tye mesh in the current code is generated by an ellitic method as a whole, which meets difficulty in the mesh orthogonality in the wall boundary region, which affects the accuracy in calculating the outer layer eddy viscosity coefficient. A twolayer H-tye mesh is used in [2], where an inner algebraic mesh is surrounded by an outer ellitic method generated mesh. he the inner mesh is designed to achieve better orthogonality. hese grid generation techniques will be imlemented in the code in the next ste research work. Unsteady searated flow simulation he searation is believed to bring high unsteadiness to the cascade flow attern. he inlet Mach number is an imortant factor which affects the searation characteristics [7]. o study the influence of the inlet Mach number on the unsteady characteristics of the searated flow, numerical simulation is carried out for high incidence angle cases with Mach number 0.5, 0.8 and.8. Each unsteady calculation is carried out based on its corresonding steady state result. Due to the limitation of the comutation caability, the hysical time interval is chosen as large as 0% of the characteristic time of the cascade, t c c U in. U in is the inlet velocity. his time interval varies with the inlet Mach number. he CFL number used in the seudo time Gauss-Seidel imlicit iteration is wenty seudo time stes are used for each hysical time ste. wo arameters are recorded to analyze the unsteady characteristics of the searation flow. he first is the mid-san searation bubble length(x), which is marked by the streamwise zero velocity oint at the first inner mesh oint on the suction surface. he second arameter is the unsteady static ressure () measured at the location of 3% downstream the leading edge on the suction surface, which is the same as the exeriment measurement location. he unsteady ressure is refered as check oint ressure in the following. Ma = 0.5 In the case of Ma = 0.5, the hysical time interval is set as ms. Fig. 3 shows the time history of the searation length oscillation in a time segment of 6 ms (30t c ). he searation length increases raidly from 0.45c to 0.66c in the first.63 ms (3t c ) and then decreases to 0.63c at t 2 ms (3.8t c ). he searation region then grows u again toward downstream to 0.73c at t 3 ms (5.7t c ). With the time rogressing, the searation region boundary oscillates back and forth on the suction surface. he average length tends to increase gradually until a eriodic state is reached after 4.4 ms (8.3t c ). he oscillation of the searation length is between 0.73c and 0.76c with a fixed cycle. he eriodicity information is clearly extracted using the FF technique. he searation region oscillation sectra is calculated from the unsteady data after 4.4 ms (8.3t c ). he frequency sectrum is shown in Fig. 4, which clearly shows a eak at 770 Hz. his indicates the searation length oscillates with a eriod of.25 ms (2.37t c ). Comared with the steady state solution, the unsteady searation calculation results in larger searation size. he reason is not clear. he unsteady check oint ressure data shows similar characteristics of the unsteady searation flow. Fig. 5 shows a segment of 6 ms (30t c ) ressure oscillation data. he ressure start at from the steady state results. he oscillation is between 2.68 and 2.72 after t 4.4 ms (8.3t c ). he oscilla- tion amlitude is about.5% of the averaged ressure level. he frequency sectrum is shown in Fig. 6 with a eak at 770 Hz. he mechanism behind the unsteady characteristics of the searation is illustrated in Fig. 8, where the evolution of a searation oscillation cycle is lotted. he stream lines at 8 time stes show the leading edge vortex shedding develoment. here are 4 hysical time stes (0.2 ms, 0.4t c ) between 2 sequential lots. he relationshi between the oscillation of the ressure and the searation length is shown in Fig. 7. At the starting time level a (t ms, 8.4t c ), the searation region has just assed the maximum length location. here are 2 vortexes in the searation bubble. hey are rotating in the same direction. he check oint ressure is going u. At time level b (t ms, 8.8t c ), the two vortexes are ushed toward downstream. he second vortex diminishes. he first vor- tex grows quickly and becomes the only vortex in the searation bubble. he searation region becomes thicker in the san-wise direction, but shorter in the stream-wise direction. he surface check oint ressure reaches its maximum level at this time level. At time level c (t ms, 9.2t c ), the searation bubble aroaches its shortest length in stream-wise direction, and maxi- mum thickness in the san-wise direction. he check oint res- 6 Coyright c 2004 by ASME

7 sure is going down. At time level d (t ms, 9.6t c ), the searation bubble has assed the minimum length location, and begins to extend toward downstream. he check oint ressure is still going down. At time level e (t ms, 0.0t c ), a new vortex is generated at leading edge and becomes the first vortex. he check oint ressure reaches its minimum value. he searation length is still increasing. At time level f (t=4854 ms, 0.4t c ), the first vortex continues to grow. he second vortex is ushed toward downstream. Both the check oint ressure and the searation length are going u. he latter is aroaching its maximum location. At time level g (t ms, 0.8t c ), the two vortexes have almost the same size, the flow structure is close to the starting time level a. he searation boundary has assed its maximum location and begin to shrink toward ustream. he check oint ressure is going u. At time level h (t ms,.2t c ), the second vortex diminishes. A new cycle is started at this time level. he leading edge vortex shedding exhibits obvious eriodical attern in its evolution rocess. he leading edge kees generating new vortexes. he new vortex ushes the old vortex bubble toward downstream and the old vortex decreases in size at the same time. When the two vortexes become of the same size, the maximum searation length is reached, where the searation bubble has the thinnest size in san-wise direction. As the new vortex grows further, the old vortex will diminish. he searation bubble will move ustream and makes the bubble thicker. he leading edge surface check oint ressure reaches its maximum level when the searation bubble shrinks and reaches its minimum level when the searation bubble boundary extends. he vortex generation, ressure variation and searation length oscillation have the same frequency characteristics with a hase difference as shown in Fig. 7. Such oscillation is maybe one of the reasons to cause to flutter. Ma = 0.8 he hysical time interval used in the calculation for the case of Mach number 0.8 is ms. A similar flowfield unsteady characteristics is exhibited in the comutation results. Fig. 9 and Fig. 2 show the searation length and the checkoint static ressure oscillation history in a time eriod of 2 ms. A clear eriodicity is shown in these two figures. It is found in the time averaged steady state study in [7] that, the increase of the inlet Mach number will enlarge the searation bubble in size. Fig. 2 indicates that the inlet Mach number increase also increases the amlitude of the ressure oscillation. he oscillation amlitude is increased to about 5% of the averaged ressure level. he increased kinetic energy in the inflow bring higher unsteadiness intensity to the searated flow filed. he corresonding FF frequency analysis is shown in Fig. 20 and Fig. 22 resectively. he frequency analysis is based on the oscillation data after t=5 ms. he unsteady searation flow exhibits higher oscillation frequency because of the increased inlet Mach number. A clear frequency sectrum eak is shown at 400 Hz in both figures, twice the frequency in the case of Ma = 0.5. Ma =.8 In the steady state simulation of the cascade at Ma=.8 in [7], the searation flow characteristics are very different from those at subsonic. he further increased kinetic energy in the inflow makes the flow attached to the blade surface in the leading edge. A smaller sized searation region aears after the shock wave because of the interaction of the shock wave and the turbulent boundary layer. he searation bubble shrinks in size and is located only in a small region at the center of the suction surface region. he hysical time interval in the calculation is set as ms. he ressure check oint is located outside of the searation region in the suersonic case. he ressure oscillation history is shown in Fig. 23. he oscillation amlitude is very small comared to the cases of Ma = 0.5 and Ma = 0.8. he flow tends to steady at the leading edge. he ressure oscillation frequency sectrum is shown in Fig. 24. he comuted characteristics of the searation flow above is similar to the exeriment measurement in [2]. In [2], when the blade is fixed, the blade surface ressure for low subsonic inlet flow at Ma=0.5 and low suersonic inlet flow at Ma=. exhibits very low unsteadiness and very strong self-induced oscillations with a frequency of 0Hz is observed in high subsonic inlet flow at Ma=0.8. However, the strong low frequency oscillation is attributed to the tunnel resonance characteristics instead of the flow unsteadiness due to the flow searation in [8]. Even though, the cascade flow searation is believed to have a direct relation with the wall surface ressure unsteady oscillation, which is an imortant factor to the flutter. Futher detailed numerical research is necessary to discover the mechanism. Conclusions A 3D unsteady comressible Navier-Stokes solver is develoed in this aer to numerically study the unsteady characteristics of the searation flow in a transonic flutter cascade under a high incidence angle of 0. he dual time steing method is alied to achieve the second order accuracy time marching. he linearized governing equation system is solved by the Gauss- Seidel line iteration method. he Baldwin-Lomax model is used to simulate the turbulence effects. he following conclusions are drawn from this study.. he high incidence cascade searation flow shows a sinusoidal attern on the oscillation of the surface ressure and the searation bubble size. A frequency sectrum eak is obtained at 770Hz for the case of Ma=0.5 and 400Hz for the case of Ma= Coyright c 2004 by ASME

8 2. he leading edge vortex shedding is the mechanism behind the unsteady characteristics of the subsonic high incidence searation flow. New vortexes are continuously generated at the suction surface leading edge. he new vortex grows and ushes the old vortexes downstream. he interaction between the vortexes results in the eriodical oscillation of the searation bubble size and the surface ressure. he vortex generation, ressure variation and searation length oscillation have the same frequency characteristics with a hase difference. 3. he characteristics of the searation flow is determined by the inlet Mach number. When the inlet flow goes from lower subsonic to higher subsonic, the size and the oscillation intensity of the searation bubble are enhanced. he flowfield oscillation eak frequency increases. When the inflow goes further to suersonic, the flow is attached on the leading edge. A small size searation bubble due to the interaction of the shock wave and the turbulent boundary layer is located right after the shock wave. [9] Bogar,. J., Gajben, M., and Kroutil, J. C., 98. Characteristic frequency and lengh scales in transonic diffuser flow oscillations. AIAA Paer [0] Leicovsky, J., McFarland, E. R., Chima, R. V., and Wood, J. R., On flowfield eriodicity in the nasa transonic flutter cascade, art I exerimental study. NASA/M Mar.. [] Grüber, B., and Carstens, V., 200. he imact of viscous effects on the aerodynamic daming of vibrating transonic comressor blades a numerical study. Journal of urbomachinery, 23. [2] Weber, S., and Platzer, M. F., A Navier-Stokes analysis of the stall flutter characteristics of the Buffum cascade. Journal of urbomachinery, 22. [3] Leicovsky, J., McFarland, E. R., Caece, V. R., Jett,. A., and Senyitko, R. G., Methodology of blade unsteady ressure measurement in the nasa transonic flutter cascade. NASA/M Oct.. Acknowledgement his work is artially suorted by AFOSR Grant F monitored by Dr. Fariba Fahroo. REFERENCES [] Leicovsky, J., Chima, R. V., Jett,. A., Bencic,. J., and Weiland, K. E., Investigation of flow searation in a transonic-fan linear cascade using visualization methods. NASA/M Dec.. [2] Leicovsky, J., McFarland, E. R., Caece, V. R., and Hayden, J., Unsteady ressures in a transonic fan cascade due to a single oscillating airfoil. ASME Paer G [3] Baldwin, B. S., and Lomax, H., 978. hin layer aroximation and algebraic model for searated turbulent flows. AIAA Paer [4] Leer, B. V., 977. owards the ultimate convservative difference scheme, III. Jouranl of Comutational Physics, 23, [5] Roe, P., 98. Aroximate Riemann solvers, arameter vectors, and difference schemes. Journal of Comuational Physics, 43, [6] van Leer, B., 982. Flux-vector slitting for the Euler equations. Lecture Note in Physics, 70. [7] Hu, Z., Zha, G., and Leicovsky, J., Numerical study on flow searation of a transonic cascade, Jan. AIAA Paer [8] Leicovsky, J., Caece, V. R., and Ford, C.., Resonance effects in the NASA transonic flutter cascade facility. ASME Paer G Figure. ransonic inlet diffuser comutation mesh Figure 2. ransonic inlet diffuser Mach contour 8 Coyright c 2004 by ASME

9 ressure variation stagger line i inlet flow γ θ c axial direction x y itchwise direction s ξ time (s) Figure 3. ransonic inlet diffuser outlet wall ressure variation Figure 6. Cascade Structure Parameter [3] E D Power Sectral Density Numerical Power Sectral Density Exeriment η ξ G F C 70 o 60 o Frequency (Hz) Frequency (Hz) Figure 4. ransonic inlet diffuser outlet wall ressure sectrum H inlet flow B y 70 o x A Figure 7. Cascade 2D Comutation Domain inlet flow Figure 5. est Section of the NASA ransonic Flutter Cascade [3] Figure 8. Cascade 3D Mesh 9 Coyright c 2004 by ASME

10 ! Exerimental Pressure Side Comutational Pressure Side Exerimental Suction Side Comutational Suction Side C Figure 9. Cascade 2D Mesh at 3 San-wise Planes x/l Figure 2. 0 Mid-san Surface Pressure Distribution at Incidence Angle x Figure 0. Mid-san Stream Lines at Incidence Angle t (ms) Figure 3. Searation Zone Length Variation with ime sf Figure. 0 Mid-san Suction Surface Streamlines at Incidence Angle Figure frequency (Hz) Searation Zone Length Variation Frequency Sectrum 0 Coyright c 2004 by ASME

11 "! t (ms) vertex vertex 2 ressure check oint, searation boundary location, x a, t ms, 8.4t c b, t ms, 8.8t c Figure 5. Suction Surface Check Point Pressure Variation with ime sf frequency (Hz) c, t ms, 9.2t c d, t ms, 9.6t c Figure 6. Suction Surface Check Point Pressure Frequency Sectrum x e, t ms, 0.0t c f, t ms, 0.4t c x" a b c d e f g h time ste Figure 7. Check Point Pressure and Searation Locations Relation g, t ms, 0.8t c h, t ms,.2t c Figure 8. Searation Bubble Evolution Coyright c 2004 by ASME

12 ! "! "! x sf t (ms) frequency (Hz) Figure 9. Searation Zone Length Oscillation (Ma=0.8) Figure 22. Checkoint Pressure Oscillation Frequency Sectrum (Ma=0.8) sf frequency (Hz) t (ms) Figure 20. (Ma=0.8) Searation Zone Length Oscillation Frequency Sectrum Figure 23. Checkoint Pressure Oscillation (Ma=.8) e e sf 6e-06 4e e t (ms) frequency (Hz) Figure 2. Checkoint Pressure Oscillation (Ma=0.8) Figure 24. Checkoint Pressure Oscillation Frequency Sectrum (Ma=.8) 2 Coyright c 2004 by ASME