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Acoustic resonances and trapped modes in annular plate cascades

  • WERNER KOCH (a1)


As a stepping stone towards understanding acoustic resonances in axial flow compressors, acoustic resonances are computed numerically in fixed single and tandem plate cascades in an infinitely long annular duct. Applying perfectly matched layer absorbing boundary conditions in the form of the complex scaling method of atomic and molecular physics to approximate the radiation condition the resonance problem is transformed into an eigenvalue problem. Of particular interest are resonances with zero radiation damping (trapped modes) or very small radiation damping (nearly trapped modes). Such resonances can be excited by wakes from compressor cascades or struts. If the shedding frequency is sufficiently close to an acoustic resonant frequency, the latter may control the vortex shedding causing high-intensity tonal noise or occasionally even blade failure. All resonances are computed for zero mean flow approximating low-Mach-number flows. The influence of various cascade parameters on the resonant frequencies is studied and, whenever possible, our numerical results are compared with published experimental findings.


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Abramowitz, M. & Stegun, I. 1965 Handbook of Mathematical Functions. Dover.
Aguilar, J. & Combes, J. 1971 A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269279.
Baslev, E. & Combes, J. 1971 Spectral properties of many body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280294.
Béerenger, J. 1994 A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185200.
Berry, M. & Wilkinson, M. 1984 Diabolical points in the spectra of triangles. Proc. R. Soc. Lond. A 392 (1802), 1543.
Callan, M., Linton, C. & Evans, D. 1991 Trapped modes in two-dimensional waveguides. J. Fluid Mech. 229, 5164.
Camp, T. 1999 A study of acoustic resonance in a low-speed multistage compressor. ASME J. Turbomach. 121, 3643.
Chew, W. & Weedon, W. 1994 A 3-D perfectly matched medium from modified Maxwell's equation with stretched coordinates. Microw. Opt. Technol. Lett. 7 (13), 599604.
Collino, F. & Monk, P. 1998 The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (6), 20612090.
Cooper, A. & Peake, N. 2006 Rotor-stator interaction noise in swirling flow: stator sweep and lean effects. AIAA J. 44 (5), 981991.
Cumpsty, N. & Whitehead, D. 1971 The excitation of acoustic resonances by vortex shedding. J. Sound Vib. 18, 353369.
Duan, Y. 2004 Trapped modes and acoustic resonances. PhD thesis, Loughborough University, Loughborough.
Duan, Y., Koch, W., Linton, C. & McIver, M. 2007 Complex resonances and trapped modes in ducted domains. J. Fluid Mech. 571, 119147.
Duan, Y. & McIver, M. 2004 Rotational acoustic resonances in cylindrical waveguides. Wave Motion 39, 261274.
Duclos, P. & Exner, P. 1995 Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1), 73102.
Elhadidi, B. & Atassi, H. 2005 Passive control for turbofan tonal noise. AIAA J. 43 (11), 22792292.
Envia, E. & Nallasamy, M. 1999 Design selection and analysis of a swept and leaned stator concept. J. Sound Vib. 228 (4), 793836.
Evans, D. 1992 Trapped acoustic modes. IMA J. Appl. Math. 49 (1), 4560.
Evans, D., Levitin, M. & Vassiliev, D. 1994 Existence theorems for trapped modes. J. Fluid Mech. 261, 2131.
Evans, D. & Linton, C. 1991 Trapped modes in open channels. J. Fluid Mech. 225, 153175.
Evans, D. & Linton, C. 1994 Acoustic resonance in ducts. J. Sound Vib. 173, 8594.
Fric, T., Villarreal, R., Auer, R., James, M., Ozgur, D. & Staley, T. 1998 Vortex shedding from struts in annular exhaust diffuser. ASME J. Turbomach. 120, 186192.
Harari, I., Patlashenko, I. & Givoli, D. 1998 Dirichlet-to-Neumann maps for unbounded wave guides. J. Comput. Phys. 143, 200223.
Hein, S., Hohage, T. & Koch, W. 2004 On resonances in open systems. J. Fluid Mech. 506, 255284.
Hein, S., Hohage, T., Koch, W. & Schöberl, J. 2007 Acoustic resonances in a high lift configuration. J. Fluid Mech. 582, 179202.
Hein, S. & Koch, W. 2008 Acoustic resonances and trapped modes in pipes and tunnels. J. Fluid Mech. 605, 401428.
Hellmich, B. 2008 Acoustic resonance in a high-speed axial compressor. PhD thesis, Universität Hannover, Hannover.
Hellmich, B. & Seume, J. 2008 Causes of acoustic resonances in a high-speed axial compressor. ASME J. Turbomach. 130, 031003-1031003-9.
Hislop, P. & Sigal, I. 1996 Introduction to Spectral Theory. Springer.
Hu, F. 2004 Absorbing boundary conditions. Intl J. Comput. Fluid Dyn. 18 (6), 513522.
Johnson, C. & Loehrke, R. 1984 An experimental investigation of wake edge tones. AIAA J. 22, 12491253.
Kim, S. & Pasciak, J. 2009 The computation of resonances in open systems using a perfectly matched layer. Math. Comp. S0025–5718(09)02227-3.
Koch, W. 1983 Resonant acoustic frequencies of flat plate cascades. J. Sound Vib. 88, 233242.
Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA J. 43 (11), 23422349.
Lamoureux, P. & Weaver, D. 1991 The effects of turbulence and damping on pipeline acoustic resonance. Proc. Inst. Mech. Engrs., Part C 205, 303312.
Legerton, M. 1992 Flow induced acoustic resonances. PhD thesis, University of Wales, Swansea.
Legerton, M., Stoneman, S. & Parker, R. 1991 An experimental investigation into flow induced acoustic resonances in an annular cascade. Proc. Inst. Mech. Engrs., Part C 205, 445452.
Li, Y. & Mei, C. 2006 Subharmonic resonance of a trapped wave near a vertical cylinder in a channel. J. Fluid Mech. 561, 391416.
Linton, C. & McIver, P. 1998 Acoustic resonances in the presence of radial fins in circular cylindrical waveguides. Wave Motion 28, 99117.
Linton, C. & McIver, M. 2002 Periodic structures in waveguides. Proc. R. Soc. Lond. A 458, 30033021.
Linton, C. & Ratcliffe, K. 2004 Bound states in coupled guides. I. Two dimensions. J. Math. Phys. 45 (4), 13591379.
Maniar, H. & Newman, J. 1997 Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309330.
Mohany, A. & Ziada, S. 2005 Flow-excited acoustic resonance of two tandem cylinders in cross-flow. J. Fluids Struct. 21 (1), 103119.
Moiseyev, N. 1998 Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 211293.
Nayfeh, A. & Huddleston, D. 1979 Resonant acoustic frequencies of parallel plates (AIAA–Paper 79–1522). In Twelfth AIAA Fluid and Plasma Dynamics Conference, Williamsburg, VA.
Parker, R. 1966 Resonance effects in wake shedding from parallel plates: some experimental observations. J. Sound Vib. 4 (1), 6272.
Parker, R. 1967 a Resonance effects in wake shedding from compressor blading. J. Sound Vib. 6, 302309.
Parker, R. 1967 b Resonance effects in wake shedding from parallel plates: calculation of resonant frequencies. J. Sound Vib. 5, 330343.
Parker, R. 1968 An investigation of acoustic resonance effects in an axial flow compressor stage. J. Sound Vib. 8, 281297.
Parker, R. 1984 Acoustic resonances and blade vibration in axial flow compressors. J. Sound Vib. 92, 529539.
Parker, R. 1997 Aeroacoustics. Intl J. Fluid Dyn. 1, (, Article 1.
Parker, R. & Griffiths, W. 1968 Low frequency resonance effects in wake shedding from parallel plates. J. Sound Vib. 7, 371379.
Parker, R. & Pryce, D. 1974 Wake excited resonances in an annular cascade: an experimental investigation. J. Sound Vib. 37, 247261.
Parker, R. & Stoneman, S. 1985 An experimental investigation of the generation and consequences of acoustic waves in an axial-flow compressor: large axial spacing between blade rows. J. Sound Vib. 99 (2), 169182.
Parker, R. & Stoneman, S. 1987 An experimental investigation of the generation and consequences of acoustic waves in an axial-flow compressor: the effect of variations in the axial spacing between blade rows. J. Sound Vib. 116 (3), 509525.
Parker, R. & Stoneman, S. 1989 The excitation and consequences of acoustic resonances in enclosed fluid flow around solid bodies. Proc. Inst. Mech. Engrs 203, 919.
Porter, R. 2007 Trapped modes in thin elastic plates. Wave Motion 45, 315.
Porter, R. & Evans, D. 1999 Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386, 233258.
Ragab, S. & Salem-Said, A.-H. 2007 Response of a flat-plate cascade to incident vortical waves. AIAA J. 45 (9), 21402148.
Rockwell, D. 1983 Oscillations of impinging shear layers. AIAA J. 21 (5), 645664.
Schöberl, J. 1997 NETGEN: an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1, 4152.
Schulten, J. 1997 Vane sweep effects on rotor/stator interaction noise. AIAA J. 35 (6), 945951.
Simon, B. 1973 The theory of resonances for dilation analytic potentials and the foundations of time dependent perturbation theory. Ann. Math. 97, 247274.
Stoneman, S., Hourigan, K., Stokes, A. & Welsh, M. 1988 Resonant sound caused by flow past two plates in tandem in a duct. J. Fluid Mech. 192, 455484.
Tyler, J. & Sofrin, T. 1962 Axial compressor noise studies. Soc. Automot. Engrs Trans. 70, 309332.
Ulbricht, I. 2002 Stabilität des stehenden Ringgitters. PhD thesis, Technische Universität Berlin, Berlin.
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347358.
Utsunomiya, T. & Eatock Taylor, R. 1999 Trapped modes around a row of circular cylinders in a channel. J. Fluid Mech. 386, 259279.
Woodley, B. & Peake, N. 1999 a Resonant acoustic frequencies of a tandem cascade. Part 1: zero relative motion. J. Fluid Mech. 393, 215240.
Woodley, B. & Peake, N. 1999 b Resonant acoustic frequencies of a tandem cascade. Part 2: rotating blade rows. J. Fluid Mech. 393, 241256.
Woodward, R., Elliott, D., Hughes, C. & Berton, J. 2001 Benefits of swept-and-leaned stators for fan noise reduction. J. Aircr. 38 (6), 11301138.
Ziada, S., Oengören, A. & Vogel, A. 2002 Acoustic resonance in the inlet scroll of a turbo-compressor. J. Fluids Struct. 16 (3), 361373.
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Acoustic resonances and trapped modes in annular plate cascades

  • WERNER KOCH (a1)


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