Sound Transmission Through a Cascade

We consider now the scattering of an incident acoustic mode of m = —8 and n = 0 and compare that to the scattering of a vortical disturbance. We take the upwash component to be the same at the mean radius and equal to unity for both disturbances. For the vortical disturbance the upwash is constant along the radius but for the acoustic mode it changes slightly (about 10%). This upwash corresponds to an acoustic mode with a coefficient c_8,o = 1.18. In the two cases, Mo = 0.3536, Mq = 0.1, Mr = 0.1, rh/rt = 0.6667, c/rm = 0.3491

at the mean radius, Cj = 3n, and a grid of {nx x n x nr} = {161 x 21 x 21} is used. The vortical disturbance has Ur = 0.

Figure (4) compares the magnitude of the unsteady lift coefficient along the span for the two cases. The lift coefficient is much higher for the acous­tic disturbance. This may be explained by the fact that the wavelength of the acoustic wave in the x-direction is much larger than that of the vortical wave. This causes phase cancellation along the blade in the case of the vortical dis­turbance thus reducing the lift coefficient compared to the case of upstream acoustic disturbance.

The magnitude of the upstream and downstream acoustic coefficients for the two cases are compared in Table (2). The acoustic coefficients are much higher in the case of the acoustic disturbance especially for the first acoustic mode.

Table 2. Magnitude of the upstream and downstream acoustic coefficients cmn for acoustic and vortical incident disturbances. Vortical Upstream Disturbance Acoustic Upstream Disturbance First Downstream Mode 0.1471 0.8608 Second Downstream Mode 0.0908 0.1458 First Upstream Mode 0.0185 0.0718 Second Upstream Mode 0.0498 0.0775

This is because the incident acoustic wave is the same as the first propagating duct mode and hence has identical radial profile.

Comparing the upstream and downstream coefficients in the case of acoustic disturbance, we see that most of the acoustic energy of the incident acoustic waves has been transmitted and propagated downstream and that the part which is refected is relatively small.

2. Conclusions

Numerical results are presented to examine the effect of mean swirl on the aerodynamic and acoustic coefficients of an annular cascade. The results sug­gest that the swirl introduces additional nonuniformities which modify the physics of the scattering in three major ways: (i) it modifies the number of propagating acoustic modes in the duct, (ii) it changes their radial variation in the duct, and (iii) it causes significant amplitude and radial phase variation of the incident disturbances.

Acknowledgments

This research was supported by The Office of Naval Research grant No. N00014-00-1-0130 with Dr. Ki-Han Kim as program manager. H. M. Atassi would like to thank the Ohio Aerospace Institute Aeroacoustic Consortium for its support.

References

Ali, A. A. (2001). Aeroacoustics and stability of Swirling Flow. PhD thesis, University of Notre Dame.

Ali, A. A., Atassi, O. V., and Atassi, H. M. (2000). Acoustic eigenmodes in a coannular duct with a general swirling fbw. AIAA 2000-1954, 6th AIAA/CEAS Aeroacoustics Conference at Maui, Hawaii.

Ali, A. A., Atassi, O. V., and Atassi, H. M. (2001). Derivation and implementation of in – fow/outfow conditions for aeroacoustic problems with swirling ft>ws. In AIAA 2001-2173, 7th AIAA/CEAS Aeroacoustics Conference at Maastricht, The Netherlands.

Atassi, H. M. (1994). Unsteady Aerodynamics of Vortical flbws: early and recent developments.

In Aerodynamics and Aeroacoustics, pages 121-172. ed. Fung, K.-Y., World Scientific.

Atassi, H. M., Ali, A. A., and Atassi, O. (2004). Scattering of incident disturbances by an annu­lar cascade in a swirling few. Journal of Fluid Mechanics, to appear.

Atassi, O. V. and Ali, A. A. (2002). Infew/outfew conditions for time-harmonic internal aeroa – coustic problems. Journal of Computational Acoustics, 10(2).

Elhadidi, B. M., Atassi, H. M., Envia, E., and Podboy, G. (2000). Evolution of rotor wake in swirling fbw. AIAA 2000-1991, 6th AIAA/CEAS Aeroacoustics Conference at Maui, Hawaii.

Glegg, S. A. L. (1999). The response of a swept blade row to a three dimensional gust. Journal of Sound and Vibration, 227(1):29-64.

Goldstein, M. E. (1976). Aeroacoustics. Mcgraw-Hill International Book Company.

Golubev, V. V. and Atassi, H. M. (1998). Acoustic-vorticity waves in swirling fews. Journal of Sound and Vibration, 209(2):203-222.

Golubev, V. V. and Atassi, H. M. (2000a). Interaction of unsteady swirling flows with annular cascades. part II. aerodynamic blade response. AIAA Journal, 38:1150-1157.

Golubev, V. V. and Atassi, H. M. (2000b). Unsteady swirling fews in annular cascades. part I. evolution of incident disturbances. AIAA Journal, 38:1142-1149.

Hamad, G. and Atassi, H. M. (1981). Sound generation in a cascade by 3D disturbance con-
vected in a subsonic few. In AIAA 81-2046, 7th Aeroacoustic Conference, Palo Alto, CA.

Kaji, S. and Okazaki, T. (1970a). Generation of sound by rotor-stator interaction. Journal of Sound and Vibration, 13:281-307.

Kaji, S. and Okazaki, T. (1970b). Propagation of sound waves through a blade row. II. Analysis
based on the acceleration potential method. Journal of Sound and Vibration, 11:355-375.

Kerrobrock, J. L. (1977). Small disturbances in turbomachine annuli with swirl. AIAA Journal, 15:794-803.

Montgomery, M. D. and Verdon, J. M. (1997). A three-dimensional linearized unsteady Euler analysis for turbomachinery blade rows. NASA CR-4770.

Namba, M. (1987). Three-dimensional few. In Unsteady Turbomachinery Aerodynamics, AGARD – AG-298.

Namba, M. and Schulten, J. B. H. M. (2000). Category 4-fan stator with harmonic excitation by rotor wake. NASA/CP-2000-209790.

Peake, N. and Kerschen, E. J. (1995). A uniform asymptotic approximation for high-frequency unsteady cascade few. Proc. R. Soc. Lond., A 449:177-186.

Podboy, G. G., Krupar, M. J., Christopher, E. H., and Richard, P. W. (2002). Fan noise source diagnostic test-ldv measured few field results. In AIAA 2002-2431, 8th AIAA/CEAS Aeroa­coustics Conference at Breckenridge, Colorado.

Schulten, J. B. H. M. (1982). Sound generated by rotor wakes interacting with a leaned vane stator. AIAA Journal, 20:1352-1358.

Smith, S. N. (1971). Discrete frequency sound generation in axial fews of turbomachines. CUED/A-Turbo/TR29.

Tyler, J. M. and Sofrin, T. G. (1962). Axial few compressor noise studies. SAE Transactions, 70:309-332.

Ventres, C. S. (1980). A computer program to calculate cascade 2D kernel. NASA Technical Memorandum.