Numerical applications

A coupling calculation is performed on the same blade at a higher rotation speed for two inter-blade phase angles, o0 = 0° and or = 360°/22. The only changes in the aerodynamic conditions are: rotation speed ^ = 4516.8 rpm, pressure ratio P2/Рц = 1.12. For the inlet ft>w, the Mach number is 0.5, the velocity is 166 m/s and the mass ft>w is 487.95 kg/s. Like in the previous case, the coupled simulations are performed under transonic conditions.

A first coupling calculation is performed for an inter-blade phase angle oо = 0°. The projection basis is formed by the first two bending modes and the

first torsion modes whose frequencies are respectively 101.7 Hz (1F), 212.8 Hz (2F) and 448.9 Hz (1T). The generalized unsteady aerodynamic forces are computed at the frequencies given in Table 3.

Figure 4 shows the flitter diagrams obtained with the double scanning and the smoothing methods. Both methods give similar results : the blade is stable in the upstream infinite velocity range from 100 m/s to 250 m/s.

A time simulation is performed for an upstream infinite velocity = 166 m/s with the minimum state smoothing method. The aerodynamic forces are approximated using 6 poles with a relative error of 1.13% . The time incre­ment is determined to have 60 time steps per period (third mode). Figure 5 shows the time evolution of the three generalized coordinates q i, q2 and q3. Table 4 shows a Fourier analysis of those signals. The resulting frequencies and damping factors are similar to those obtained from the computation in the frequency domain (double scanning method).

Table 3 Case 2 – Excitation frequencies Phase angle ст0 = 0° Frequencies f (Hz) 0 5.0 70.0 101.7 230.1 448.9 600.0 Reduced frequencies к = 2% f/Voc 0 0.185 2.59 3.755 8.5 16.58 22.16 Phase angle cti = 360° /22 Frequencies f (Hz) 10.0 20.0 50.0 105.3 139 191.8 278.3 350.0 Reduced frequencies к = 2% f/Voc 0.37 0.74 1.85 3.89 5.26 7.08 10.28 12.93

w» Bow (Vys)

Figure 6 Case 2, crі – Flutter diagram

Figure 5 Case 2, <ro – Time evolution ol q,

Table 4 Case 2, сто – Frequencies (Hz) and Table 5 Case 2, a1 – Frequencies (Hz) and damping factors (V% = 166 m/s) damping factors (V% = 166 m/s) Smoothing (time) Double scanning Smoothing (time) Double scanning Freq. Damp. Freq. Damp. Freq. Damp. Freq. Damp. <?1 101.02 0.0080 100.96 0.0080 <?i 104.92 0.0092 104.14 0.0091 <12 212.18 0.0062 211.86 0.0063 <12 191.34 0.0031 191.42 0.0034 <13 447.00 0.0069 447.21 0.0073 <13 277.40 0.0022 277.73 0.0023

A coupling calculation is now performed for an inter-blade phase angle a1 = 1 x 360°/22. The method is then tested for a non-zero phase angle and for a complex projection basis. The latter is formed by the first two bending modes and the first torsion mode whose frequencies are respectively 105 Hz (1F), 191.8 Hz (2F) and 278.3 Hz (1T). The generalized unsteady aerodynamic forces are computed for the frequencies given in Table 3.

Figure 6 shows the flitter diagram resulting from a computation using the double scanning method. As in the zero phase angle case, the blade is stable in the upstream infinite frequency range from 100 m/s to 250 m/s.

A time calculation is performed for an upstream infinite velocity = 166 m/s. The time increment is determined to have 60 time steps per period (third mode). The smoothing method uses 8 poles and gives a relative error of 1.18%. As for a zero phase angle, the frequencies and the damping factors resulting from the computations in the time domain (smoothing method) and in the frequency domain (double scanning method) are similar (Table 5).

A non-linear friction force (model of Coulomb) is applied on the blade for the phase angle a0. The coupled simulation is performed by projecting the motion equation of the structure on a Craig and Bampton basis which is com­posed of six normal modes and one static constraint modes, and by using the minimum state smoothing of the generalized aerodynamics forces in the time domain. Figure 7 shows the time evolution of the generalized coordinates q 1 (first normal mode) and q7 (static mode, on which the friction force is applied) for different intensities of the friction force: 0 N (linear case), 50 N and 500 N. We can see the infflence of the intensity of the friction force on the response.

2. Conclusion

Two fliid-structure coupling methods for a rotating bladed disk system based on the cyclic symmetry properties and the projection on the complex modes are presented. The double scanning method is more robust but it is only applica­ble for solving the flutter equation in the frequency domain. The minimum state smoothing method can be used for both frequency and time domain sim­ulations but its implementation is more complex and it requires more attention and know-how from the user. The main advantage of the proposed indirect cou­pling method in the time domain is that it is generally less time consuming than the direct coupling method since the aerodynamic calculations are performed only once at the beginning of the simulation and not at each time step. There­fore, the aerodynamic forces do not need to be computed again if the modes that generate them are unchanged and they can be re-used for other computa­tions, for example when the applied external forces change or when we want to study the influence of a friction damper on the stability of the bladed-disk. However, the proposed indirect coupling method is more restrictive and less accurate since it is based on the assumptions of linearized aerodynamics and harmonic motion, which is not the case of the direct coupling method.

The good agreement between the results of the different methods shows that the extension of the proposed coupling methods from aircraft applications to turbomachinary is feasible. However, further validation tests should be carried out under various conditions such as other rotation speeds, phase angles, pres­sure ratios, viscous fluid etc. The future work consists in numerical simulations of an unstable configuration and the introduction of the blade mistuning.

References

Craig, Jr., R. R. and Bampton, M. C. C. (1968). Coupling of substructures for dynamic analysis. AIAA Journal, 6(7):1313-1319.

Dat, R. and Meurzec, J.-L. (1969). Sur les calculs de fbttementpar lamethode dite du “balayage en frequence reduite”. La Recherche Aerospatiale, (133):41-43.

Dugeai, A., Madec, A., and Sens, A. S. (2000). Numerical unsteady aerodynamics for turbo­machinery aeroelasticity. In Proceedings of the 9th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, Lyon, pages 830-840. Karpel, M. (1982). Design for active flitter suppression and gust alleviation using state-space aeroelastic modeling. AIAA Journal of Aircraft, 19(3):221-227.

Poirion, F. (1995). Modelisation temporelle des systemes aeroservoelastiques. Application a l’etude des effets des retards. La Recherche Aerospatiale, (2):103—114.

Sayma, A. I., Vahdati, M., and Imregun, M. (2000). An integrated nonlinear approach for turbo­machinery forced response prediction. Part I: Formulation. Journal of Fluids and Structures, 14:87—101.

Tran, D.-M., Labaste, C., and Liauzun, C. (2003). Methods of fliid-structure coupling in fre­quency and time domain using linearized aerodynamics for turbomachinery. Journal of Flu­ids and Structures, 17(8):1161-1180.

The simulations were performed for the following aerodynamic conditions: rotation speed U = 4066.4 rpm, upstream total temperature Tц = 288 K, upstream total pressure Рц = 101325 Pa, pressure ratio P2/Рц = 1.05, phase angle do = 0 °. The steady aerodynamic simulation gives for the inflow a Mach number of 0.5, a velocity of 166 m/s and a mass ft>w of 462.92 kg/s. Figure 2 shows a shock whose position depends on the applied pressure ratio. The coupled simulations are then performed under transonic conditions.

For the coupling computation, only the first two bending modes of the blade whose frequencies are 97.4 Hz (1F) and 204.3 Hz (2F) are retained to form the projection basis. Unsteady aerodynamic simulations are performed using the mode shapes as oscillating motion shape for the frequencies given in Table 1.

In order to perform a time simulation, a minimum state smoothing of the generalized aerodynamic forces is performed with a relative error of 0.197%. The time increment is determined to have 60 time steps per period (second mode). An initial velocity is applied to all generalized coordinates. Figure 3 shows the time evolution of both generalized coordinates computed using three

methods : the first method is the indirect coupling in the time domain (smooth­ing method), the second one is a direct coupling numerical simulation using a grid deformation technique, and the last one is a direct coupling numerical simulation using a blowing condition. The results from the three methods are similar, although the aeroelastic damping computed with the direct simulation using the grid deformation technique is quite a lot smaller than those com­puted with the other methods. The good agreement between the results of the smoothing method and those of the direct coupling method with a blowing con­dition can be explained by the fact that a blowing condition was also used in the smoothing method. Table 2 gives the frequencies and the damping factors computed using the three methods. Moreover, the results from the smooth­ing method in the time domain is similar to the one from the double scanning method in the frequency domain. The time simulation has been performed for an upstream infinite velocity = 166 m/s.

(Зеге>а1’1«к! COOfJrwile Qt

Figure 3 Case 1 – Time evolution of q1 and q2

Table 2 Case 1 – Frequencies (Hz) and damping factors of q1 and q2 (VTO = 166 m/s) Time domain Frequency domain Smoothing Direct (grid deform.) Direct (blowing) Double scanning Freq. Damping Freq. Damping Freq. Damping Frequency Damping q[ 96.70 0.0082 96.94 0.0061 96.54 0.0077 96.69 0.0085 ~q2 203.21 0.0056 203.47 0.0025 203.54 0.0048 203.13 0.0057

The previously described coupling methods have been applied to a numeri­cal model of a compressor disk composed of 22 large chord blades, for differ­ent rotation speeds and phase angles. The structural finite element model of a reference sector with one blade has 19 539 degrees of freedom. The eigenfre – quencies and modes in vacuum are computed by using the cyclic symmetry and
by taking into account the stiffening and softening rotational effects. The aero­dynamic computations were performed on a two-block structured grid, each block having 61x18x30 points with 60 points on the profile. Figure 1 shows the mesh of one channel of the embedding Arid. For the coupling calculation, the blade is assumed not to have any structural damping. In addition to the indirect coupling methods described in this paper, and for comparison, we also use the direct coupling method in the time domain (Case 1).