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Nonlinear dynamics and acoustic radiation of coherent structures consisting of multiple ring–helical modes in the near-nozzle region of a subsonic turbulent circular jet

Published online by Cambridge University Press:  11 October 2023

Zhongyu Zhang Open the ORCID record for Zhongyu Zhang [Opens in a new window]  and
Xuesong Wu Open the ORCID record for Xuesong Wu [Opens in a new window]
Zhongyu Zhang
Affiliation:
Laboratory of High-Speed Aerodynamics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, PR China
Xuesong Wu*
Affiliation:
School of Mechanical Engineering, Nantong University, Nantong 226019, PR China Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, UK
*
†Email address for correspondence: x.wu@imperial.ac.uk
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Abstract

This paper investigates the nonlinear evolution and acoustic radiation of coherent structures (CS) in the near-nozzle region of a subsonic turbulent circular jet. A CS is taken to be a wavepacket consisting of multiple ring/helical modes, which are considered to be inviscid instability waves supported by the mean-flow profile. As the three-dimensionality of helical modes is weak in the near-nozzle region, the ring and helical modes with the same frequency have nearly the same growth rates and critical levels. They coexist and interact with each other in their common critical layer at high Reynolds numbers. The self and mutual quadratic interactions generate a mean-flow distortion and streaks, which act back on the fundamental components through the cubic interaction. The amplitude of the CS is governed by an integro-partial-differential equation, a significant feature of which is that differentiations with respect to the azimuthal coordinate appear in the history-dependent nonlinear terms. The non-parallelism of the mean flow as well as the impact of fine-scale turbulence on CS are taken into account and found to affect the nonlinear terms. By solving the amplitude equation, the development of the constituting modes, streamwise vortices and streaks are described. For CS consisting of frequency sideband, low-frequency components are excited nonlinearly and amplify to reach a considerable level. By analysing the large-distance asymptote of the perturbation, the low-frequency acoustic waves are found to be emitted by the temporally–spatially varying mean-flow distortion and streaks generated by the nonlinear interactions of the CS, and are thereby determined on the basis of first principles. Interestingly, the energetic part of the streaky structure that contributes to the nonlinear dynamics does not radiate directly, and instead the Reynolds stresses driving the subdominant radiating components represent the true physical sources.

JFM classification

Acoustics: Jet noise Instability: Nonlinear instability Waves/Free-surface Flows: Critical layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 973 , 25 October 2023 , A8
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Alkislar, M.B., Krothapalli, A. & Butler, G.W. 2007 The effect of streamwise vortices on the aeroacoustics of a Mach 0.9 jet. J. Fluid Mech. 578, 139–169.CrossRefGoogle Scholar
Arndt, R.E.A., Long, D.F. & Glauser, M.N. 1997 The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J. Fluid Mech. 340, 1–33.CrossRefGoogle Scholar
Batchelor, G.K. & Gill, A.E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (4), 529–551.CrossRefGoogle Scholar
Bechert, D. & Pfizenmaier, E. 1975 On the amplification of broad band jet noise by a pure tone excitation. J. Sound Vib. 43 (3), 581–587.CrossRefGoogle Scholar
Brès, G.A., Jaunet, V., Le Rallic, M., Jordan, P., Towne, A., Schmidt, O., Colonius, T., Cavalieri, A.V.G. & Lele, S.K. 2016 Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound. In 22nd AIAA/CEAS Aeroacoustics Conference, Lyon, France, AIAA paper 2016–3050. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
Brès, G.A., Jordan, P., Jaunet, V., Rallic, M.L., Cavalieri, A.V.G., Towne, A., Lele, S.K., Colonius, T. & Schmidt, O.T. 2018 Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. 851, 83–124.CrossRefGoogle Scholar
Bridges, J.E. & Hussain, A.K.M.F. 1987 Roles of initial condition and vortex pairing in jet noise. J. Sound Vib. 117 (2), 289–311.CrossRefGoogle Scholar
Bridges, J. & Hussain, F. 1992 Direct evaluation of aeroacoustic theory in a jet. J. Fluid Mech. 240, 469–501.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P., Agarwal, A. & Gervais, Y. 2011 Jittering wave-packet models for subsonic jet noise. J. Sound Vib. 330 (18–19), 4474–4492.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P., Colonius, T. & Gervais, Y. 2012 Axisymmetric superdirectivity in subsonic jets. J. Fluid Mech. 704, 388–420.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
Cavalieri, A.V.G., Rodriguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559–592.CrossRefGoogle Scholar
Churilov, S.M. & Shukhman, I.G. 1994 Nonlinear spatial evolution of helical disturbances to an axial jet. J. Fluid Mech. 281, 371–402.CrossRefGoogle Scholar
Citriniti, J.H. & George, W.K. 2000 Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137–166.CrossRefGoogle Scholar
Cohen, J. & Wygnanski, I. 1987 a The evolution of instabilities in the axisymmetric jet. Part 1. The linear growth of disturbances near the nozzle. J. Fluid Mech. 176, 191–219.CrossRefGoogle Scholar
Cohen, J. & Wygnanski, I. 1987 b The evolution of instabilities in the axisymmetric jet. Part 2. The flow resulting from the interaction between two waves. J. Fluid Mech. 176, 221–235.CrossRefGoogle Scholar
Corke, T.C. & Kusek, S.M. 1993 Resonance in axisymmetric jets with controlled helical-mode input. J. Fluid Mech. 249, 307–336.CrossRefGoogle Scholar
Crawley, M., Gefen, L., Kuo, C.W., Samimy, M. & Camussi, R. 2018 Vortex dynamics and sound emission in excited high-speed jets. J. Fluid Mech. 839, 313–347.CrossRefGoogle Scholar
Crow, S.C. 1970 Aerodynamic sound emission as a singular perturbation problem. Stud. Appl. Maths 49 (1), 21–46.CrossRefGoogle Scholar
Crow, S.C. & Champagne, F.H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 547–591.CrossRefGoogle Scholar
Davoust, S., Jacquin, L. & Leclaire, B. 2012 Dynamics of $m = 0$ and $m = 1$ modes and of streamwise vortices in a turbulent axisymmetric mixing layer. J. Fluid Mech. 709, 408–444.CrossRefGoogle Scholar
Duan, L., Choudhari, M.M. & Wu, M. 2014 Numerical study of acoustic radiation due to a supersonic turbulent boundary layer. J. Fluid Mech. 746, 165–192.CrossRefGoogle Scholar
Ffowcs Williams, J.E. & Kempton, A.J. 1978 The noise from the large-scale structure of a jet. J. Fluid Mech. 84, 673–694.CrossRefGoogle Scholar
Fuchs, H.V. & Michel, U. 1978 Experimental evidence of turbulent source coherence affecting jet noise. AIAA J. 16 (9), 871–872.CrossRefGoogle Scholar
Goldstein, M.E. 2001 An exact form of Lilley's equation with a velocity quadrupole/temperature dipole source term. J. Fluid Mech. 443, 231–236.CrossRefGoogle Scholar
Goldstein, M.E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315–333.CrossRefGoogle Scholar
Goldstein, M.E. & Leib, S.J. 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96.CrossRefGoogle Scholar
Haberman, R. 1972 Critical layers in parallel shear flows. Stud. Appl. Maths 51, 139–161.CrossRefGoogle Scholar
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, 277–307.CrossRefGoogle Scholar
Hussain, A.K.M.F. & Hasan, M.A.Z. 1985 Turbulence suppression in free turbulent shear flows under controlled excitation. Part 2. Jet-noise reduction. J. Fluid Mech. 150, 159–168.CrossRefGoogle Scholar
Hussain, A.K.M.F. & Reynolds, W.C. 1972 The mechanics of an organized wave in turbulent shear flow. Part 2. Experimental results. J. Fluid Mech. 54, 241–261.CrossRefGoogle Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195.CrossRefGoogle Scholar
Jung, D., Gamard, S. & George, W.K. 2004 Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 1. The near-field region. J. Fluid Mech. 514, 173–204.CrossRefGoogle Scholar
Juvé, D., Sunyach, M. & Comte-Bellot, G. 1979 Filtered azimuthal correlations in the acoustic far field of a subsonic jet. AIAA J. 17 (1), 112–113.CrossRefGoogle Scholar
Juvé, D., Sunyach, M. & Comte-Bellot, G. 1980 Intermittency of the noise emission in subsonic cold jets. J. Sound Vib. 71 (3), 319–332.CrossRefGoogle Scholar
Kantharaju, J., Courtier, R., Leclaire, B. & Jacquin, L. 2020 Interactions of large-scale structures in the near field of round jets at high Reynolds numbers. J. Fluid Mech. 888, A8.CrossRefGoogle Scholar
Kearney-Fischer, M., Sinha, A. & Samimy, M. 2013 Intermittent nature of subsonic jet noise. AIAA J. 51 (5), 1142–1155.CrossRefGoogle Scholar
Kœnig, M., Sasaki, K., Cavalieri, A.V.G., Jordan, P. & Gervais, Y. 2016 Jet-noise control by fluidic injection from a rotating plug: linear and nonlinear sound-source mechanisms. J. Fluid Mech. 788, 358–380.CrossRefGoogle Scholar
Laufer, J. 1964 Some statistical properties of the pressure field radiated by a turbulent boundary layer. Phys. Fluids 7 (8), 1191–1197.CrossRefGoogle Scholar
Leib, S.J. 1991 Nonlinear evolution of subsonic and supersonic disturbances on a compressible free shear layer. J. Fluid Mech. 224, 551–578.CrossRefGoogle Scholar
Leib, S.J. & Lee, S.S. 1995 Nonlinear evolution of a pair of oblique instability waves in a supersonic boundary layer. J. Fluid Mech. 282, 339–371.CrossRefGoogle Scholar
Liepmann, D. & Gharib, M. 1992 The role of streamwise vorticity in the near-field entrainment of round jets. J. Fluid Mech. 245, 643–668.CrossRefGoogle Scholar
Lighthill, M.J. 1952 On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. A 211, 564–587.Google Scholar
Lilley, G.M. 1974 On the noise from jets. In Noise Mechanisms, AGARD-CP-131, pp. 13.1–13.12.Google Scholar
Liu, J.T.C. 1974 Developing large-scale wavelike eddies and the near jet noise field. J. Fluid Mech. 62, 437–464.CrossRefGoogle Scholar
Long, T.A. & Petersen, R.A. 1992 Controlled interactions in a forced axisymmetric jet. Part 1. The distortion of the mean flow. J. Fluid Mech. 235, 37–55.CrossRefGoogle Scholar
Lorteau, M., Cléro, F. & Vuillot, F. 2015 Analysis of noise radiation mechanisms in hot subsonic jet from a validated large eddy simulation solution. Phys. Fluids 27 (7), 075108.CrossRefGoogle Scholar
Mair, M., Bacic, M., Chakravarthy, K. & Williams, B. 2020 Jet preferred mode vs shear layer mode. Phys. Fluids 32 (6), 064106.CrossRefGoogle Scholar
Mattingly, G.E. & Chang, C.C. 1974 Unstable waves on an axisymmetric jet column. J. Fluid Mech. 65 (3), 541–560.CrossRefGoogle Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199.CrossRefGoogle Scholar
Michalke, A. & Fuchs, H.V. 1975 On turbulence and noise of an axisymmetric shear flow. J. Fluid Mech. 70, 179–205.CrossRefGoogle Scholar
Miksad, R.W. 1973 Experiments on nonlinear interactions in the transition of a free shear layer. J. Fluid Mech. 59, 1–21.CrossRefGoogle Scholar
Moore, C.J. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80, 321–367.CrossRefGoogle Scholar
Narayanan, S., Barber, T.J. & Polak, D.R. 2002 High subsonic jet experiments: turbulence and noise generation studies. AIAA J. 40 (3), 430–437.CrossRefGoogle Scholar
Nogueira, P.A.S., Cavalieri, A.V.G., Jordan, P. & Jaunet, V. 2019 Large-scale streaky structures in turbulent jets. J. Fluid Mech. 873, 211–237.CrossRefGoogle Scholar
Paschereit, C.O., Oster, D., Long, T.A., Fiedler, H.E. & Wygnanski, I. 1992 Flow visualization of interactions among large coherent structures in an axisymmetric jet. Exp. Fluids 12 (3), 189–199.CrossRefGoogle Scholar
Pickering, E., Rigas, G., Nogueira, P.A.S., Cavalieri, A.V.G., Schmidt, O.T. & Colonius, T. 2020 Lift-up, Kelvin–Helmholtz and Orr mechanisms in turbulent jets. J. Fluid Mech. 896, A2.CrossRefGoogle Scholar
Plaschko, P. 1979 Helical instabilities of slowly divergent jets. J. Fluid Mech. 92 (2), 209–215.CrossRefGoogle Scholar
Ronneberger, D. & Ackermann, U. 1979 Experiments on sound radiation due to non-linear interaction of instability waves in a turbulent jet. J. Sound Vib. 62 (1), 121–129.CrossRefGoogle Scholar
Sandham, N.D. & Salgado, A.M. 2008 Nonlinear interaction model of subsonic jet noise. Phil. Trans. R. Soc. Lond. A 366 (1876), 2745–2760.Google ScholarPubMed
Sasaki, K., Cavalieri, A.V.G., Jordan, P., Schmidt, O.T., Colonius, T. & Brès, G.A. 2017 High-frequency wavepackets in turbulent jets. J. Fluid Mech. 830, R2.CrossRefGoogle Scholar
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, 1153–1181.CrossRefGoogle Scholar
Schmidt, O.T., Towne, A., Rigas, G., Colonius, T. & Brès, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.CrossRefGoogle Scholar
Sinha, A., Rodríguez, D., Brès, G.A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 71–95.CrossRefGoogle Scholar
Sparks, C.A. & Wu, X. 2008 Nonlinear development of subsonic modes on compressible mixing layers: a unified strongly nonlinear critical-layer theory. J. Fluid Mech. 614, 105–144.CrossRefGoogle Scholar
Strange, P.J.R. & Crighton, D.G. 1983 Spinning modes on axisymmetric jets. Part 1. J. Fluid Mech. 134, 231–345.CrossRefGoogle Scholar
Suponitsky, V., Sandham, N.D. & Morfey, C.L. 2010 Linear and nonlinear mechanisms of sound radiation by instability waves in subsonic jets. J. Fluid Mech. 658, 509–538.CrossRefGoogle Scholar
Tam, C.K.W. 2019 A phenomenological approach to jet noise: the two-source model. Phil. Trans. R. Soc. Lond. A 377 (2159), 20190078.Google ScholarPubMed
Tam, C.K.W. & Burton, D.E. 1984 Sound generated by instability waves of supersonic flow. Part 2. Axisymmetric jets. J. Fluid Mech. 138, 273–295.CrossRefGoogle Scholar
Tam, C., Golebiowski, M. & Seiner, J. 1996 On the two components of turbulent mixing noise from supersonic jets. In 2nd AIAA/CEAS Aeroacoustics Conference, State College, PA, USA, AIAA Paper 1996-1716. American Institute of Aeronautics and Astronautics.CrossRefGoogle Scholar
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, 349–381.CrossRefGoogle Scholar
Tam, C.K.W., Viswanathan, K., Ahuja, K.K. & Panda, J. 2008 The sources of jet noise: experimental evidence. J. Fluid Mech. 615, 253–292.CrossRefGoogle Scholar
Tinney, C.E., Glauser, M.N. & Ukeiley, L.S. 2008 a Low-dimensional characteristics of a transonic jet. Part 1. Proper orthogonal decomposition. J. Fluid Mech. 612, 107–141.CrossRefGoogle Scholar
Tinney, C.E., Ukeiley, L.S. & Glauser, M.N. 2008 b Low-dimensional characteristics of a transonic jet. Part 2. Estimate and far-field prediction. J. Fluid Mech. 615, 53–92.CrossRefGoogle Scholar
Tso, J. & Hussain, F. 1989 Organized motions in a fully developed turbulent axisymmetric jet. J. Fluid Mech. 203, 425–448.CrossRefGoogle Scholar
Uzun, A. & Hussaini, M.Y. 2009 Simulation of noise generation in the near-nozzle region of a chevron nozzle jet. AIAA J. 47 (8), 1793–1810.CrossRefGoogle Scholar
Viswanathan, K. 2004 Aeroacoustics of hot jets. J. Fluid Mech. 516, 39–82.CrossRefGoogle Scholar
Viswanathan, K. 2008 Investigation of noise source mechanisms in subsonic jets. AIAA J. 46 (8), 2020–2032.CrossRefGoogle Scholar
Wan, Z., Yang, H., Zhang, X. & Sun, D. 2016 Instability waves and aerodynamic noise in a subsonic transitional turbulent jet. Eur. J. Mech. B/Fluids 57, 192–203.CrossRefGoogle Scholar
Wu, X. 1993 Nonlinear temporal–spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices. J. Fluid Mech. 256, 685–719.CrossRefGoogle Scholar
Wu, X. 2005 Mach wave radiation of nonlinearly evolving supersonic instability modes in shear layers. J. Fluid Mech. 523, 121–159.CrossRefGoogle Scholar
Wu, X. 2011 On generation of sound in wall-bounded shear flows: back action of sound and global acoustic coupling. J. Fluid Mech. 689, 279–316.CrossRefGoogle Scholar
Wu, X. 2019 Nonlinear theories for shear-flow instabilities: physical insights and practical implications. Annu. Rev. Fluid Mech. 51, 421–485.CrossRefGoogle Scholar
Wu, X. & Hogg, L.W. 2006 Acoustic radiation of Tollmien–Schlichting waves as they undergo rapid distortion. J. Fluid Mech. 550, 307–347.CrossRefGoogle Scholar
Wu, X. & Huerre, P. 2009 Low-frequency sound radiated by a nonlinear modulated wavepacket of helical modes on a subsonic circular jet. J. Fluid Mech. 637, 173–211.CrossRefGoogle Scholar
Wu, X., Lee, S.S. & Cowley, S.J. 1993 On the weakly nonlinear three-dimensional instability of shear layers to pairs of oblique waves: the Stokes layer as a paradigm. J. Fluid Mech. 253, 681–720.CrossRefGoogle Scholar
Wu, X. & Tian, F. 2012 Spectral broadening and flow randomization in free shear layers. J. Fluid Mech. 706, 431–469.CrossRefGoogle Scholar
Wu, X. & Zhang, Z. 2019 First-principle description of acoustic radiation of shear flows. Phil. Trans. R. Soc. Lond. A 377 (2159), 20190077.Google ScholarPubMed
Wu, X. & Zhuang, X. 2016 Nonlinear dynamics of large-scale coherent structures in turbulent free shear layers. J. Fluid Mech. 787, 396–439.CrossRefGoogle Scholar
Zhang, Z. 2022 Roll & roar of large eddies: nonlinear dynamics and low-frequency acoustic radiation of coherent structures on subsonic turbulent free shear layers. PhD thesis, Tianjin University.Google Scholar
Zhang, P., Wan, Z. & Sun, D. 2021 The influence of nonlinearities on jet noise modeling based on parabolized stability equation. Phys. Fluids 33 (8), 086107.CrossRefGoogle Scholar
Zhang, Z. & Wu, X. 2020 Nonlinear evolution and acoustic radiation of coherent structures in subsonic turbulent free shear layers (referred to as ‘[I]’). J. Fluid Mech. 884, A10.CrossRefGoogle Scholar
Zhang, Z. & Wu, X. 2022 Nonlinear evolution and low-frequency acoustic radiation of ring-mode coherent structures on subsonic turbulent circular jets (referred to as ‘[II]’). J. Fluid Mech. 940, A39.CrossRefGoogle Scholar
Zhang, Z. & Wu, X. 2023 a Generation of sound waves by nonlinearly evolving ring-mode coherent structures on a turbulent subsonic circular jet: a comparative study of two mechanisms. Acta Mechanica Sin. 39, 322272.CrossRefGoogle Scholar
Zhang, Z. & Wu, X. 2023 b A unified theory for the envelope radiation of ring-mode coherent structures in the very-near-nozzle and developed regions of a circular jet. Phys. Fluids 35 (1), 014113.CrossRefGoogle Scholar