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Acoustic resonance in the potential core of subsonic jets

  • Aaron Towne (a1), André V. G. Cavalieri (a2), Peter Jordan (a3), Tim Colonius (a1), Oliver Schmidt (a1), Vincent Jaunet (a3) and Guillaume A. Brès (a4)...

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number, $0.82 .

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Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA.

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Balakumar, P.1998 Prediction of supersonic jet noise. AIAA Paper 1998-1057.
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (04), 529551.
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers I. Springer Science & Business Media.
Bers, A. 1983 Space-time evolution of plasma instabilities-absolute and convective. In Basic Plasma Physics, vol. 1. North-Holland.
Bodony, D. J. & Lele, S. K. 2008 Current status of jet noise predictions using large-eddy simulation. AIAA J. 46, 346380.
Bogey, C. & Bailly, C. 2010 Influence of nozzle-exit boundary-layer conditions on the flow and acoustic fields of initially laminar jets. J. Fluid Mech. 663, 507538.
Bogey, C., Mardsen, O. & Bailly, C. 2012 Influence of initial turbulence level on the flow and sound fields of a subsonic jet at a diameter-based Reynolds number of 105 . J. Fluid Mech. 701, 352385.
Brès, G. A., Ham, F. E., Nichols, J. W. & Lele, S. K. 2017 Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 11641184.
Brès, G. A., Jaunet, J., Le Rallic, M., Jordan, P., Colonius, T. & Lele, S. K.2015 Large eddy simulation for jet noise: the importance of getting the boundary layer right. AIAA Paper 2015-2535.
Brès, G. A., Jaunet, V., Le Rallic, M., Jordan, P., Towne, A., Schmidt, O. T., Colonius, T., Cavalieri, A. V. G. & Lele, S. K.2016 Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound. AIAA Paper 2016-3050.
Brès, G. A., Jordan, P., Colonius, T., Rallic, M. Le, Jaunet, V. & Lele, S. K.2014 Large eddy simulation of a turbulent mach 0.9 jet. Tech. Rep. Proceedings of the Center for Turbulence Research Summer Program, Stanford University.
Briggs, R. J. 1964 Electron-Stream Interactions with Plasmas. MIT.
Cavalieri, A. V. G., Rodríguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559592.
Chan, Y. Y. 1974 Spatial waves in turbulent jets. Part ii. Phys. Fluids 17 (9), 16671670.
Coleman, T. F. & Li, Y. 1996 An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418445.
Colonius, T. & Ran, H. 2002 A super-grid-scale model for simulating compressible flow on unbounded domains. J. Comput. Phys. 182 (1), 191212.
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397413.
Fontaine, R. A., Elliott, G. S., Austin, J. M. & Freund, J. B. 2015 Very near-nozzle shear-layer turbulence and jet noise. J. Fluid Mech. 770, 2751.
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 Modal and transient dynamics of jet flows. Phys. Fluids 25 (4), 044103.
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97128.
Hall, B. C. 2013 Quantum Theory for Mathematicians. Springer.
Hileman, J. I., Thurow, B. S., Caraballo, E. J. & Samimy, M. 2005 Large-scale structure evolution and sound emission in high-speed jets: real-time visualization with simultaneous acoustic measurements. J. Fluid Mech. 544, 277307.
Huerre, P. 2002 Open shear flow instabilities. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), chap. 4, pp. 159229. Cambridge University Press.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Jaunet, V., Jordan, P., Cavalieri, A. V. G., Towne, A., Colonius, T., Schmidt, O. & Brès, G. A.2016 Tonal dynamics and sound in subsonic turbulent jets. AIAA Paper 2016-3016.
Jendoubi, S. & Strykowski, P. J. 1994 Absolute and convective instability of axisymmetric jets with external flow. Phys. Fluids 6 (9), 30003009.
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173195.
Lawrence, J. & Self, R. H.2015 Installed jet-flap impingement tonal noise. AIAA Paper 2015-3118.
Lessen, M., Fox, J. A. & Zien, H. M. 1965 The instability of inviscid jets and wakes in compressible fluid. J. Fluid Mech. 21, 129143.
Lesshafft, L. & Huerre, P. 2007 Linear impulse response in hot round jets. Phys. Fluids 19 (2), 024102.
Liu, J. 1974 Developing large-scale wavelike eddies and the near jet noise field. J. Fluid Mech. 62 (03), 437464.
Lorteau, M., Cléro, F. & Vuillot, F. 2015 Analysis of noise radiation mechanisms in a hot subsonic jet from a validated large eddy simulation solution. Phys. Fluids 27, 075108.
Michalke, A.1970 A note on the saptial jet-instability of the compressible cylindrical wortex sheet. DLR Report FB-70-51.
Michalke, A.1971 Instabilitat eines kompressiblen runden freistrahls unter beriicksichtigung.
Mohseni, K. & Colonius, T. 2000 Numerical treatment of polar coordinate singularities. J. Comput. Phys. 157 (2), 787795.
Monkewitz, P. A., Huerre, P. & Chomaz, J.-M. 1993 Global linear stability analysis of weakly non-parallel shear flows. J. Fluid Mech. 251, 120.
Monkewitz, P. A. & Sohn, K. 1988 Absolute instability in hot jets. AIAA J. 26 (8), 911916.
Nichols, J. W. & Lele, S. K. 2011 Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225241.
Rienstra, S. W. 2000 Cut-on, cut-off transition of sound in slowly varying flow ducts. J. Associazione Italiana di Aeronautica e Astronautica AIDAA 79, 9396.
Rienstra, S. W. 2003 Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. J. Fluid Mech. 495, 157173.
Schmidt, O. T., Towne, A., Colonius, T., Cavalieri, A. V. G., Jordan, P. & Brès, G. A. 2017 Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. J. Fluid Mech. 825, 11531181.
Sinha, A.2011 Development of reduced-order modes and strategies for feedback control of high-speed axisymmetric jets. PhD thesis, Ohio State University.
Sinha, A., Rodriguez, D., Brès, G. A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 7195.
Strand, B. 1994 Summation by parts for finite difference approximations for d/dx . J. Comput. Phys. 110, 4767.
Suzuki, T. & Colonius, T. 2006 Instability waves in a subsonic round jet detected using a near-field phased microphone array. J. Fluid Mech. 565, 197226.
Tam, C. K. W. 1971 Directional acoustic radiation from a supersonic jet generated by shear layer instability. J. Fluid Mech. 46, 757768.
Tam, C. K. W. & Burton, D. E. 1984 Sound generated by instability waves of supersonic flows. Part 2. axisymmetric jets. J. Fluid Mech. 138, 273295.
Tam, C. K. W. & Hu, F. Q. 1989 On the three families of instability waves of high-speed jets. J. Fluid Mech. 201, 447483.
Tam, C. K. W. & Morris, P. J. 1980 The radiation of sound by the instability waves of a compressible plane turbulent shear layer. J. Fluid Mech. 98, 349381.
Thompson, K. W. 1987 Time dependent boundary conditions for hyperbolic systems. J. Comput. Phys. 68, 124.
Viswanathan, K. 2004 Aeroacoustics of hot jets. J. Fluid Mech. 516, 3982.
Yen, C. C. & Messersmith, N. L. 1998 Application of parabolized stability equations to the prediction of jet instabilities. AIAA J. 36 (8), 15411544.
Zaman, K. B. M. Q., Fagan, A. F., Bridges, J. E. & Brown, C. A.2015 Investigating the feedback path in a jet-surface resonant interaction. AIAA Paper 2015-2999.
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