0
Review Article

Wave-Packet Models for Jet Dynamics and Sound Radiation

[+] Author and Article Information
André V. G. Cavalieri

Divisão de Engenharia Aeroespacial,
Instituto Tecnológico de Aeronáutica,
São José dos Campos, SP 12228-900, Brazil
e-mail: andre@ita.br

Peter Jordan

Chargé de Recherche,
Institut Pprime,
Dép. Fluides, Thermique, Combustion,
CNRS—Université de Poitiers—ENSMA,
Poitiers 86000, France
e-mail: peter.jordan@univ-poitiers.fr

Lutz Lesshafft

Chargé de Recherche,
Laboratoire d'Hydrodynamique (LadHyX),
CNRS—École Polytechnique,
Palaiseau 91120, France
e-mail: lutz@ladhyx.polytechnique.fr

Manuscript received November 1, 2018; final manuscript received January 31, 2019; published online March 13, 2019. Editor: Harry Dankowicz.

Appl. Mech. Rev 71(2), 020802 (Mar 13, 2019) (27 pages) Paper No: AMR-18-1137; doi: 10.1115/1.4042736 History: Received November 01, 2018; Revised January 31, 2019

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Cavalieri, A. V. G. , Rodríguez, D. , Jordan, P. , Colonius, T. , and Gervais, Y. , 2013, “ Wavepackets in the Velocity Field of Turbulent Jets,” J. Fluid Mech., 730(9), pp. 559–592. [CrossRef]
Fuchs, H. V. , and Michel, U. , 1978, “ Experimental Evidence of Turbulent Source Coherence Affecting Jet Noise,” AIAA J., 16(9), pp. 871–872. [CrossRef]
Juvé, D. , Sunyach, M. , and Comte-Bellot, G. , 1979, “ Filtered Azimuthal Correlations in the Acoustic Far Field of a Subsonic Jet,” AIAA J., 17, p. 112. [CrossRef]
Kopiev, V. , Zaitsev, M. , Velichko, S. , Kotova, A. , and Belyaev, I. , 2008, “ Cross-Correlations of Far Field Azimuthal Modes in Subsonic Jet Noise,” AIAA Paper No. 2008-2887.
Cavalieri, A. V. G. , Jordan, P. , Colonius, T. , and Gervais, Y. , 2012, “ Axisymmetric Superdirectivity in Subsonic Jets,” J. Fluid Mech., 704, p. 388. [CrossRef]
Faranosov, G. A. , Belyaev, I. V. , Kopiev, V. F. , Zaytsev, M. Y. , Aleksentsev, A. A. , Bersenev, Y. V. , Chursin, V. A. , and Viskova, T. A. , 2016, “ Adaptation of the Azimuthal Decomposition Technique to Jet Noise Measurements in Full-Scale Tests,” AIAA J., 55(2), pp. 572–584. [CrossRef]
Brès, G. A. , Jordan, P. , Jaunet, V. , Le Rallic, M. , Cavalieri, A. V. , Towne, A. , Lele, S. K. , Colonius, T. , and Schmidt, O. T. , 2018, “ Importance of the Nozzle-Exit Boundary-Layer State in Subsonic Turbulent Jets,” J. Fluid Mech., 851, pp. 83–124. [CrossRef]
Brès, G. A. , Ham, F. E. , Nichols, J. W. , and Lele, S. K. , 2017, “ Unstructured Large-Eddy Simulations of Supersonic Jets,” AIAA J., 55(4), pp. 1164–1184. [CrossRef]
Mollo-Christensen, E. , 1967, “ Jet Noise and Shear Flow Instability Seen From an Experimenter's Viewpoint (Similarity Laws for Jet Noise and Shear Flow Instability as Suggested by Experiments),” ASME J. Appl. Mech., 34(1), pp. 1–7. [CrossRef]
Lau, J. C. , Fisher, M. J. , and Fuchs, H. V. , 1972, “ The Intrinsic Structure of Turbulent Jets,” J. Sound Vib., 22(4), pp. 379–384. [CrossRef]
Michalke, A. , and Fuchs, H. V. , 1975, “ On Turbulence and Noise of an Axisymmetric Shear Flow,” J. Fluid Mech., 70(1), pp. 179–205. [CrossRef]
Armstrong, R. R. , Fuchs, H. V. , and Michalke, A. , 1977, “ Coherent Structures in Jet Turbulence and Noise,” AIAA J., 15(7), pp. 1011–1017. [CrossRef]
Tinney, C. E. , and Jordan, P. , 2008, “ The Near Pressure Field of Co-Axial Subsonic Jets,” J. Fluid Mech., 611, pp. 175–204. [CrossRef]
Breakey, D. E. , Jordan, P. , Cavalieri, A. V. , Nogueira, P. A. , Léon, O. , Colonius, T. , and Rodríguez, D. , 2017, “ Experimental Study of Turbulent-Jet Wave Packets and Their Acoustic Efficiency,” Phys. Rev. Fluids, 2(12), p. 124601. [CrossRef]
Jung, D. , Gamard, S. , and 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, pp. 173–204. [CrossRef]
Suzuki, T. , and Colonius, T. , 2006, “ Instability Waves in a Subsonic Round Jet Detected Using a Near-Field Phased Microphone Array,” J. Fluid Mech., 565, pp. 197–226. [CrossRef]
Jaunet, V. , Jordan, P. , and Cavalieri, A. , 2017, “ Two-Point Coherence of Wave Packets in Turbulent Jets,” Phys. Rev. Fluids, 2(2), p. 024604. [CrossRef]
Crow, S. C. , and Champagne, F. H. , 1971, “ Orderly Structure in Jet Turbulence,” J. Fluid Mech., 48(3), pp. 547–591. [CrossRef]
Moore, C. J. , 1977, “ The Role of Shear-Layer Instability Waves in Jet Exhaust Noise,” J. Fluid Mech., 80(2), pp. 321–367. [CrossRef]
Michalke, A. , 1971, “ Instabilitat Eines Kompressiblen Runden Freistrahls Unter Berucksichtigung Des Einflusses Der Strahlgrenzschichtdicke,” Z. Flugwiss., 19, pp. 319–328.
Crighton, D. G. , and Gaster, M. , 1976, “ Stability of Slowly Diverging Jet Flow,” J. Fluid Mech., 77(2), pp. 397–413. [CrossRef]
Sasaki, K. , Cavalieri, A. V. , Jordan, P. , Schmidt, O. T. , Colonius, T. , and Brès, G. A. , 2017, “ High-Frequency Wavepackets in Turbulent Jets,” J. Fluid Mech., 830, p. R2.
Schmid, P. J. , and Henningson, D. S. , 2001, Stability and Transition in Shear Flows, Vol. 142, Springer, New York.
Criminale, W. O. , Jackson, T. L. , and Joslin, R. D. , 2003, Theory and Computation of Hydrodynamic Stability, Cambridge University Press, Cambridge, UK.
Schmid, P. J. , 2007, “ Nonmodal Stability Theory,” Annu. Rev. Fluid Mech., 39(1), pp. 129–162. [CrossRef]
Juniper, M. P. , Hanifi, A. , and Theofilis, V. , 2014, “ Modal Stability Theory—Lecture Notes From the Flow-Nordita Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013,” ASME Appl. Mech. Rev., 66(2), p. 024804. [CrossRef]
Schmid, P. J. , and Brandt, L. , 2014, “ Analysis of Fluid Systems: Stability, Receptivity, Sensitivity,” ASME Appl. Mech. Rev., 66(2), p. 024803. [CrossRef]
Kim, J. , and Bewley, T. R. , 2007, “ A Linear Systems Approach to Flow Control,” Annu. Rev. Fluid Mech., 39(1), pp. 383–417. [CrossRef]
Bagheri, S. , Henningson, D. , Hoepffner, J. , and Schmid, P. , 2009, “ Input-Output Analysis and Control Design Applied to a Linear Model of Spatially Developing Flows,” ASME Appl. Mech. Rev., 62(2), p. 020803. [CrossRef]
Fabbiane, N. , Semeraro, O. , Bagheri, S. , and Henningson, D. S. , 2014, “ Adaptive and Model-Based Control Theory Applied to Convectively Unstable Flows,” ASME Appl. Mech. Rev., 66(6), p. 060801. [CrossRef]
Sipp, D. , and Schmid, P. J. , 2016, “ Linear Closed-Loop Control of Fluid Instabilities and Noise-Induced Perturbations: A Review of Approaches and Tools,” ASME Appl. Mech. Rev., 68(2), p. 020801. [CrossRef]
Farrell, B. F. , and Ioannou, P. J. , 1993, “ Stochastic Forcing of the Linearized Navier–Stokes Equations,” Phys. Fluids A, 5(11), pp. 2600–2609. [CrossRef]
Jovanović, M. R. , and Bamieh, B. , 2005, “ Componentwise Energy Amplification in Channel Flows,” J. Fluid Mech., 534, pp. 145–183. [CrossRef]
McKeon, B. , and Sharma, A. , 2010, “ A Critical-Layer Framework for Turbulent Pipe Flow,” J. Fluid Mech., 658, pp. 336–382. [CrossRef]
Hwang, Y. , and Cossu, C. , 2010, “ Linear Non-Normal Energy Amplification of Harmonic and Stochastic Forcing in the Turbulent Channel Flow,” J. Fluid Mech., 664, pp. 51–73. [CrossRef]
Semeraro, O. , Jaunet, V. , Jordan, P. , Cavalieri, A. V. , and Lesshafft, L. , 2016, “ Stochastic and Harmonic Optimal Forcing in Subsonic Jets,” AIAA Paper No. 2016-2935.
Towne, A. , Schmidt, O. T. , and Colonius, T. , 2018, “ Spectral Proper Orthogonal Decomposition and Its Relationship to Dynamic Mode Decomposition and Resolvent Analysis,” J. Fluid Mech., 847, pp. 821–867. [CrossRef]
Schmidt, O. T. , Towne, A. , Rigas, G. , Colonius, T. , and Brès, G. A. , 2018, “ Spectral Analysis of Jet Turbulence,” J. Fluid Mech., 855, pp. 953–982. [CrossRef]
Garnaud, X. , Lesshafft, L. , Schmid, P. , and Huerre, P. , 2013, “ The Preferred Mode of Incompressible Jets: Linear Frequency Response Analysis,” J. Fluid Mech., 716, pp. 189–202. [CrossRef]
Jordan, P. , and Colonius, T. , 2013, “ Wave Packets and Turbulent Jet Noise,” Annu. Rev. Fluid Mech., 45(1), pp. 173–195.
Waleffe, F. , 1997, “ On a Self-Sustaining Process in Shear Flows,” Phys. Fluids, 9(4), pp. 883–900. [CrossRef]
Moehlis, J. , Faisst, H. , and Eckhardt, B. , 2004, “ A Low-Dimensional Model for Turbulent Shear Flows,” New J. Phys., 6(1), p. 56. [CrossRef]
Barkley, D. , 2016, “ Theoretical Perspective on the Route to Turbulence in a Pipe,” J. Fluid Mech., 803, p. P1.
Lighthill, M. J. , 1952, “ On Sound Generated Aerodynamically—I: General Theory,” Proc. R. Soc. London, Ser. A, 211(1107), pp. 564–587. [CrossRef]
Crighton, D. G. , 1975, “ Basic Principles of Aerodynamic Noise Generation,” Prog. Aerosp. Sci., 16(1), pp. 31–96. [CrossRef]
Goldstein, M. E. , 2003, “ A Generalized Acoustic Analogy,” J. Fluid Mech., 488, pp. 315–333. [CrossRef]
McKeon, B. , 2017, “ The Engine Behind (Wall) Turbulence: Perspectives on Scale Interactions,” J. Fluid Mech., 817, p. P1.
Garnaud, X. , Sandberg, R. D. , and Lesshafft, L. , 2013, “ Global Response to Forcing in a Subsonic Jet: Instability Wavepackets and Acoustic Radiation,” AIAA Paper No. 2013-2232.
Jeun, J. , Nichols, J. W. , and Jovanović, M. R. , 2016, “ Input-Output Analysis of High-Speed Axisymmetric Isothermal Jet Noise,” Phys. Fluids, 28(4), p. 047101. [CrossRef]
Tissot, G. , Zhang, M. , Lajús, F. C. , Cavalieri, A. V. , and Jordan, P. , 2017, “ Sensitivity of Wavepackets in Jets to Nonlinear Effects: The Role of the Critical Layer,” J. Fluid Mech., 811, pp. 95–137.
Lesshafft, L. , Semeraro, O. , Jaunet, V. , Cavalieri, A. V. G. , and Jordan, P. , 2018, “ Resolvent-Based Modelling of Coherent Wavepackets in a Turbulent Jet,” arXiv preprint arXiv:1810.09340.
Huerre, P. , and Monkewitz, P. A. , 1990, “ Local and Global Instabilities in Spatially Developing Flows,” Annu. Rev. Fluid Mech., 22(1), pp. 473–537.
Monkewitz, P. A. , Huerre, P. , and Chomaz, J.-M. , 1993, “ Global Linear Stability Analysis of Weakly Non-Parallel Shear Flows,” J. Fluid Mech., 251(1), pp. 1–20.
Goldstein, M. E. , and Leib, S. J. , 2005, “ The Role of Instability Waves in Predicting Jet Noise,” J. Fluid Mech., 525, pp. 37–72.
Cohen, J. , and Wygnanski, I. , 1987, “ The Evolution of Instabilities in the Axisymmetric Jet—Part 1: The Linear Growth of Disturbances Near the Nozzle,” J. Fluid Mech., 176(1), pp. 191–219.
Bendat, J. , and Piersol, A. , 1986, Random Data: Analysis and Measurement Procedures, Wiley-Interscience, New York.
Berkooz, G. , Holmes, P. , and Lumley, J. , 1993, “ The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annu. Rev. Fluid Mech., 25(1), pp. 539–575.
Rowley, C. W. , and Dawson, S. T. , 2017, “ Model Reduction for Flow Analysis and Control,” Annu. Rev. Fluid Mech., 49(1), pp. 387–417.
Taira, K. , Brunton, S. L. , Dawson, S. T. , Rowley, C. W. , Colonius, T. , McKeon, B. J. , Schmidt, O. T. , Gordeyev, S. , Theofilis, V. , and Ukeiley, L. S. , 2017, “ Modal Analysis of Fluid Flows: An Overview,” AIAA J., (12), pp. 4013–4041.
Picard, C. , and Delville, J. , 2000, “ Pressure Velocity Coupling in a Subsonic Round Jet,” Int. J. Heat Fluid Flow, 21(3), pp. 359–364.
Chomaz, J. M. , 2005, “ Global Instabilities in Spatially Developing Flows: Non-Normality and Nonlinearity,” Annu. Rev. Fluid Mech., 37(1), pp. 357–392.
Farrell, B. F. , and Ioannou, P. J. , 2014, “ Statistical State Dynamics: A New Perspective on Turbulence in Shear Flow,” arXiv preprint arXiv:1412.8290.
Beneddine, S. , Sipp, D. , Arnault, A. , Dandois, J. , and Lesshafft, L. , 2016, “ Conditions for Validity of Mean Flow Stability Analysis,” J. Fluid Mech., 798, pp. 485–504.
Wu, X. , and Huerre, P. , 2009, “ Low-Frequency Sound Radiated by a Nonlinearly Modulated Wavepacket of Helical Modes on a Subsonic Circular Jet,” J. Fluid Mech., 637, pp. 173–211.
Suponitsky, V. , Sandham, N. , and Morfey, C. , 2010, “ Linear and Nonlinear Mechanisms of Sound Radiation by Instability Waves in Subsonic Jets,” J. Fluid Mech., 658, pp. 509–538.
Suponitsky, V. , Sandham, N. , and Agarwal, A. , 2011, “ On the Mach Number and Temperature Dependence of Jet Noise: Results From a Simplified Numerical Model,” J. Sound Vib., 330(17), pp. 4123–4138.
Towne, A. , Bres, G. A. , and Lele, S. K. , 2017, “ A Statistical Jet-Noise Model Based on the Resolvent Framework,” AIAA Paper No. 2017-3706.
Mollo-Christensen, E. , 1963, “Measurements of Near Field Pressure of Subsonic Jets,” Advisory Group for Aeronautical Research and Development, Paris, France, Report No. 449.
Mollo-Christensen, E. , Kolpin, M. A. , and Martucelli, J. R. , 1964, “ Experiments on Jet Flows and Jet Noise Far-Field Spectra and Directivity Patterns,” J. Fluid Mech., 18(2), pp. 285–301.
Crow, S. C. , 1972, “ Acoustic Gain of a Turbulent Jet,” Meeting of Division of Fluid Dynamics, American Physical Society, University of Colorado, Boulder, CO, Nov., Paper No. IE.6.
Ffowcs Williams, J. E. , and Kempton, A. J. , 1978, “ The Noise From the Large-Scale Structure of a Jet,” J. Fluid Mech., 84(4), pp. 673–694.
Crighton, D. G. , and Huerre, P. , 1990, “ Shear-Layer Pressure Fluctuations and Superdirective Acoustic Sources,” J. Fluid Mech., 220(1), pp. 355–368.
Cavalieri, A. V. G. , Jordan, P. , Agarwal, A. , and Gervais, Y. , 2011, “ Jittering Wave-Packet Models for Subsonic Jet Noise,” J. Sound Vib., 330(18–19), pp. 4474–4492.
Cavalieri, A. V. G. , and Agarwal, A. , 2014, “ Coherence Decay and Its Impact on Sound Radiation by Wavepackets,” J. Fluid Mech., 748, pp. 399–415.
Ffowcs Williams, J. E. , 1963, “ The Noise From Turbulence Convected at High Speed,” Philos. Trans. R. Soc. London, Ser. A, 255(1061), pp. 469–503.
Freund, J. B. , 2001, “ Noise Sources in a Low-Reynolds-Number Turbulent Jet at Mach 0.9,” J. Fluid Mech., 438, pp. 277–305.
Cabana, M. , Fortuné, V. , and Jordan, P. , 2008, “ Identifying the Radiating Core of Lighthill's Source Term,” Theor. Comput. Fluid Dyn., 22(2), pp. 87–106.
Baqui, Y. B. , Agarwal, A. , Cavalieri, A. V. , and Sinayoko, S. , 2015, “ A Coherence-Matched Linear Source Mechanism for Subsonic Jet Noise,” J. Fluid Mech., 776, pp. 235–267.
Nogueira, P. A. , Cavalieri, A. V. , and Jordan, P. , 2017, “ A Model Problem for Sound Radiation by an Installed Jet,” J. Sound Vib., 391, pp. 95–115.
Kopiev, V. , Chernyshev, S. , Faranosov, G. , Zaitsev, M. , and Belyaev, I. , 2010, “ Correlations of Jet Noise Azimuthal Components and Their Role in Source Identification,” AIAA Paper No. 2010-4018.
Reba, R. , Narayanan, S. , and Colonius, T. , 2010, “ Wave-Packet Models for Large-Scale Mixing Noise,” Int. J. Aeroacoustics, 9(4–5), pp. 533–558.
Suzuki, T. , 2013, “ Coherent Noise Sources of a Subsonic Round Jet Investigated Using Hydrodynamic and Acoustic Phased-Microphone Arrays,” J. Fluid Mech., 730, pp. 659–698.
Freund, J. B. , 2003, “ Noise-Source Turbulence Statistics and the Noise From a Mach 0.9 Jet,” Phys. Fluids, 15(6), pp. 1788–1799.
Morris, P. J. , 2009, “ A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets,” Int. J. Aeroacoustics, 8(4), pp. 301–315.
Papamoschou, D. , 2018, “ Wavepacket Modeling of the Jet Noise Source,” Int. J. Aeroacoustics, 17(1–2), pp. 52–69.
McLaughlin, D. K. , Morrison, G. L. , and Troutt, T. R. , 1975, “ Experiments on the Instability Waves in a Supersonic Jet and Their Acoustic Radiation,” J. Fluid Mech., 69(1), pp. 73–95.
Papamoschou, D. , 1997, “ Mach Wave Elimination in Supersonic Jets,” AIAA J., 35(10), pp. 1604–1611.
Tam, C. K. , Chen, P. , and Seiner, J. , 1992, “ Relationship Between the Instability Waves and Noise of High-Speed Jets,” AIAA J., 30(7), pp. 1747–1752.
Tam, C. K. , 1995, “ Supersonic Jet Noise,” Annu. Rev. Fluid Mech., 27(1), pp. 17–43.
Laufer, J. , and Yen, T.-C. , 1983, “ Noise Generation by a Low-Mach-Number Jet,” J. Fluid Mech., 134(1), pp. 1–31.
Sinha, A. , Rodríguez, D. , Brès, G. A. , and Colonius, T. , 2014, “ Wavepacket Models for Supersonic Jet Noise,” J. Fluid Mech., 742, pp. 71–95.
Lighthill, M. J. , 1954, “ On Sound Generated Aerodynamically—II: Turbulence as a Source of Sound,” Proc. R. Soc. London, Ser. A, 222(1148), p. 19540049.
Freund, J. B. , and Colonius, T. , 2009, “ Turbulence and Sound-Field POD Analysis of a Turbulent Jet,” Int. J. Aeroacoustics, 8(4), pp. 337–354.
Towne, A. , Colonius, T. , Jordan, P. , Cavalieri, A. V. G. , and Brès, G. A. , 2015, “ Stochastic and Nonlinear Forcing of Wavepackets in a Mach 0.9 Jet,” AIAA Paper No. 2015-2217.
Huerre, P. , 2000, “ Open Shear Flow Instabilities,” Perspectives in Fluid Dynamics, G. Batchelor , H. Moffatt , and M. Worster , eds., Cambridge University Press, Cambridge, UK, pp. 159–229.
Lesshafft, L. , and Huerre, P. , 2007, “ Linear Impulse Response in Hot Round Jets,” Phys. Fluids, 19(2), p. 024102.
Rodríguez, D. , Cavalieri, A. V. , Colonius, T. , and Jordan, P. , 2015, “ A Study of Linear Wavepacket Models for Subsonic Turbulent Jets Using Local Eigenmode Decomposition of Piv Data,” Eur. J. Mech. B/Fluids, 49, pp. 308–321.
Monkewitz, P. A. , and Sohn, K. D. , 1988, “ Absolute Instability in Hot Jets,” AIAA J., 26(8), pp. 911–916.
Chomaz, J. , Huerre, P. , and Redekopp, L. , 1988, “ Bifurcations to Local and Global Modes in Spatially Developing Flows,” Phys. Rev. Lett., 60(1), p. 25. [PubMed]
Weideman, J. A. , and Reddy, S. C. , 2000, “ A Matlab Differentiation Matrix Suite,” ACM Trans. Math. Software, 26(4), pp. 465–519.
Lesshafft, L. , 2018, “ Artificial Eigenmodes in Truncated Flow Domains,” Theor. Comput. Fluid Dyn., 32(3), pp. 245–262.
Barkley, D. , 2006, “ Linear Analysis of the Cylinder Wake Mean Flow,” EPL (Europhys. Lett.), 75(5), p. 750.
Coenen, W. , Lesshafft, L. , Garnaud, X. , and Sevilla, A. , 2017, “ Global Instability of Low-Density Jets,” J. Fluid Mech., 820, pp. 187–207.
Chakravarthy, R. , Lesshafft, L. , and Huerre, P. , 2018, “ Global Stability of Buoyant Jets and Plumes,” J. Fluid Mech., 835, pp. 654–673.
Schmidt, O. T. , Towne, A. , Colonius, T. , Cavalieri, A. V. , Jordan, P. , and Brès, G. A. , 2017, “ Wavepackets and Trapped Acoustic Modes in a Turbulent Jet: Coherent Structure Education and Global Stability,” J. Fluid Mech., 825, pp. 1153–1181.
Trefethen, L. N. , 2000, Spectral Methods in MATLAB, Vol. 10, Society for Industrial Mathematics, Philadelphia, PA.
Tissot, G. , Lajús , F. C., Jr. , Cavalieri, A. V. , and Jordan, P. , 2017, “ Wave Packets and Orr Mechanism in Turbulent Jets,” Phys. Rev. Fluids, 2(9), p. 093901.
Colonius, T. , and Lele, S. K. , 2004, “ Computational Aeroacoustics: Progress on Nonlinear Problems of Sound Generation,” Prog. Aerosp. Sci., 40(6), pp. 345–416.
Wang, M. , Freund, J. B. , and Lele, S. K. , 2006, “ Computational Prediction of Flow-Generated Sound,” Annu. Rev. Fluid Mech., 38(1), pp. 483–512.
Freund, J. B. , 2019, “ Nozzles, Turbulence, and Jet Noise Prediction,” J. Fluid Mech., 860, pp. 1–4.
Neilsen, T. B. , Gee, K. L. , Harker, B. M. , and James, M. M. , 2016, “ Level-Educed Wavepacket Representation of Noise Radiation From a High-Performance Military Aircraft,” AIAA Paper No. 2016-1880.
Zaman, K. , Bridges, J. , and Huff, D. , 2011, “ Evolution From ‘Tabs' to ‘Chevron Technology'—A Review,” Int. J. Aeroacoustics, 10(5–6), pp. 685–709.
Henderson, B. , 2010, “ Fifty Years of Fluidic Injection for Jet Noise Reduction,” Int. J. Aeroacoustics, 9(1–2), pp. 91–122.
Shur, M. L. , Spalart, P. R. , and Strelets, M. K. , 2011, “ Les-Based Evaluation of a Microjet Noise Reduction Concept in Static and Flight Conditions,” J. Sound Vib., 330(17), pp. 4083–4097.
Kopiev, V. , and Ostrikov, N. , 2012, “ Axisymmetrical Instability Wave Control Due to Resonance Coupling of Azimuthal Modes in High-Speed Jet Issuing From Corrugated Nozzle,” AIAA Paper No. 2012-2144.
Lajús, F. C. , Cavalieri, A. V. , and Deschamps, C. J. , 2015, “ Spatial Stability Characteristics of Non-Circular Jets,” AIAA Paper No. 2015-2537.
Sinha, A. , Xia, H. , and Colonius, T. , 2016, “ Parabolized Stability Analysis of Jets Issuing From Serrated Nozzles,” Fluid-Structure-Sound Interactions and Control, Springer, Berlin, Germany, pp. 211–215.
Kœnig, M. , Sasaki, K. , Cavalieri, A. V. G. , Jordan, P. , and Gervais, Y. , 2016, “ Jet-Noise Control by Fluidic Injection From a Rotating Plug: Linear and Nonlinear Sound-Source Mechanisms,” J. Fluid Mech., 788, pp. 358–380.
Le Rallic, M. , Jordan, P. , and Gervais, Y. , 2016, “ Jet-Noise Reduction: The Effect of Azimuthal Actuation Modes,” AIAA Paper No. 2016-2868.
Sinha, A. , Towne, A. , Colonius, T. , Schlinker, R. H. , Reba, R. , Simonich, J. C. , and Shannon, D. W. , 2017, “ Active Control of Noise From Hot Supersonic Jets,” AIAA J., 56(3), pp. 933–948.
Sasaki, K. , Piantanida, S. , Cavalieri, A. V. , and Jordan, P. , 2017, “ Real-Time Modelling of Wavepackets in Turbulent Jets,” J. Fluid Mech., 821, pp. 458–481.
Beneddine, S. , Yegavian, R. , Sipp, D. , and Leclaire, B. , 2017, “ Unsteady Flow Dynamics Reconstruction From Mean Flow and Point Sensors: An Experimental Study,” J. Fluid Mech., 824, pp. 174–201.
Kopiev, V. F. , Belyaev, I. V. , Faranosov, G. A. , Kopiev, V. A. , Ostrikov, N. , Zaytsev, M. Y. , Akishev, Y. S. , Grushin, M. , Trushkin, N. , Bityurin, V. , Klimov, A. I. , Moralev, I. A. , Kossyi, I. A. , Berezhetskaya, N. K. , and Taktakishvili, M. I. , 2015, “ Instability Wave Control in Turbulent Jet by Acoustical and Plasma Actuators,” Prog. Flight Phys., 7, pp. 211–228.
Sasaki, K. , Tissot, G. , Cavalieri, A. V. , Silvestre, F. J. , Jordan, P. , and Biau, D. , 2018, “ Closed-Loop Control of a Free Shear Flow: A Framework Using the Parabolized Stability Equations,” Theor. Comput. Fluid Dyn., 32(6), pp. 765–788.
Cavalieri, A. V. G. , 2016, “ Jet-Noise Control Using Wavepacket Models,” Measurement, Simulation and Control of Subsonic and Supersonic Jet Noise ( Von Karman Institute for Fluid Dynamics Lecture Series), von Karman Institute for Fluid Dynamics, Sint-Genesius-Rode, Belgium.
Semeraro, O. , Lusseyran, F. , Pastur, L. , and Jordan, P. , 2017, “ Qualitative Dynamics of Wave Packets in Turbulent Jets,” Phys. Rev. Fluids, 2(9), p. 094605.
Cavalieri, A. V. , Jordan, P. , Wolf, W. R. , and Gervais, Y. , 2014, “ Scattering of Wavepackets by a Flat Plate in the Vicinity of a Turbulent Jet,” J. Sound Vib., 333(24), pp. 6516–6531.
Bychkov, O. , and Faranosov, G. , 2014, “ On the Possible Mechanism of the Jet Noise Intensification Near a Wing,” Acoust. Phys., 60(6), pp. 633–646.
Faranosov, G. A. , and Bychkov, O. P. , 2017, “ Two-Dimensional Model of the Interaction of a Plane Acoustic Wave With Nozzle Edge and Wing Trailing Edge,” J. Acoust. Soc. Am., 141(1), pp. 289–299. [PubMed]
Piantanida, S. , Jaunet, V. , Huber, J. , Wolf, W. R. , Jordan, P. , and Cavalieri, A. V. , 2016, “ Scattering of Turbulent-Jet Wavepackets by a Swept Trailing Edge,” J. Acoust. Soc. Am., 140(6), pp. 4350–4359. [PubMed]
Towne, A. , Cavalieri, A. V. , Jordan, P. , Colonius, T. , Schmidt, O. , Jaunet, V. , and Brès, G. A. , 2017, “ Acoustic Resonance in the Potential Core of Subsonic Jets,” J. Fluid Mech., 825, pp. 1113–1152.
Jordan, P. , Jaunet, V. , Towne, A. , Cavalieri, A. V. , Colonius, T. , Schmidt, O. , and Agarwal, A. , 2018, “ Jet–Flap Interaction Tones,” J. Fluid Mech., 853, pp. 333–358.
Lawrence, J. , and Self, R. H. , 2015, “ Installed Jet-Flap Impingement Tonal Noise,” AIAA Paper No. 2015-3118. https://arc.aiaa.org/doi/10.2514/6.2015-3118
Tam, C. K. , and Ahuja, K. , 1990, “ Theoretical Model of Discrete Tone Generation by Impinging Jets,” J. Fluid Mech., 214(1), pp. 67–87.
Bogey, C. , and Gojon, R. , 2017, “ Feedback Loop and Upwind-Propagating Waves in Ideally Expanded Supersonic Impinging Round Jets,” J. Fluid Mech., 823, pp. 562–591.
Shen, H. , and W. Tam, C. K. , 1998, “ Numerical Simulation of the Generation of Axisymmetric Mode Jet Screech Tones,” AIAA J., 36(10), pp. 1801–1807.
Edgington-Mitchell, D. , Jaunet, V. , Jordan, P. , Towne, A. , Soria, J. , and Honnery, D. , 2018, “ Upstream-Travelling Acoustic Jet Modes as a Closure Mechanism for Screech,” J. Fluid Mech., 855, R1, pp. 1–12.
Abreu, L. I. , Cavalieri, A. V. , and Wolf, W. R. , 2017, “ Coherent Hydrodynamic Waves and Trailing-Edge Noise,” AIAA Paper No. 2017-3173.

Figures

Grahic Jump Location
Fig. 1

Illustration of a Mach 0.9 jet and its sound radiation. The central part (colors in the online version) shows temperature fluctuations, highlighting turbulent disturbances, whereas the outer region (black and white in the online version) shows pressure fluctuations, which far from the jet correspond to the acoustic radiation. The right plot shows a cross section of turbulent and acoustic fields taken at x/D = 20. Figure taken from the large-eddy simulation of Brès et al. [7], using the compressible solver “Charles” [8].

Grahic Jump Location
Fig. 2

Wavepackets from the same Mach 0.9 jet simulation shown in Fig. 1: (a) sample snapshot of the axisymmetric part of the pressure field and (b) pressure from the first SPOD mode at Strouhal 0.3

Grahic Jump Location
Fig. 3

Schematic representation of resolvent analysis for a turbulent jet. FT and IFT stand for the Fourier transform and its inverse, respectively. Figure taken from Tissot et al. [50].

Grahic Jump Location
Fig. 4

Mollo-Christensen's handwritten solution of Eq. (1) for the farfield sound-pressure auto-correlation [9]

Grahic Jump Location
Fig. 5

Model time-periodic line sources: (a) compact source, khL = 0.1 and (b) extended wave-packet source, khL = 5

Grahic Jump Location
Fig. 6

Axisymmetric hydrodynamic pressure signature of a turbulent jet [14]

Grahic Jump Location
Fig. 7

Sound radiation by a time-periodic, harmonic line-source: (a) M = 0.6 and (b) M = 2.0

Grahic Jump Location
Fig. 8

Superdirectivity. Blue line: extended wavepacket model; black circles: sound pressure level of the axisymmetric mode at St = 0.2, from Ref. [5].

Grahic Jump Location
Fig. 9

Acoustic matching: (a) effect of M and khL on sound radiation, (b) effect of khL on wavepacket amplitude envelope, and (c) effect of M and khL on acoustic matching

Grahic Jump Location
Fig. 10

Axisymmetric wavepackets, at St = 0.2, educed from LES data. Top: M = 0.9 (courtesy of O. Kaplan); bottom: M = 1.5 (from Ref. [91]).

Grahic Jump Location
Fig. 11

CSD (real part) of model wavepacket sources, with khL = 5: (a) Lc = L/10, (b) Lc = L, and (c) Lc = 10L

Grahic Jump Location
Fig. 12

Sound radiation by stochastic model sources: (a) M = 0.6 and (b) M = 2.0

Grahic Jump Location
Fig. 13

Source CSD in frequency wavenumber space. Color scale is logarithmic. Wavenumbers inside the inner square (red in the online version) satisfy |ky|,|kz|/kh≤Mc for a subsonic jet (M = 0.6 and Mc = 0.36); wavenumbers inside the outer square (green in the online version) satisfy the same conditions for a supersonic jet (M = 2 and Mc = 1.2): (a) Lc = 10L and (b) Lc = L.

Grahic Jump Location
Fig. 14

Spectral proper orthogonal decomposition eigenvalues for line-source CSD, Sss: (a) POD eigenvalues and (b) amount of represented power

Grahic Jump Location
Fig. 15

Source POD modes: (a) mode 1, (b) mode 2, (c) mode 3, and (d) mode 4

Grahic Jump Location
Fig. 16

Phase of POD modes: (a) mode 1, (b) mode 2, (c) mode 3, and (d) mode 4

Grahic Jump Location
Fig. 17

Comparison of four most energetic modeled (solid lines) and measured (dashed lines) SPOD mode shapes. The first mode is shown on the bottom, and subsequent modes are plotted with the upper curves. SPOD modes taken from the measurements of Jaunet et al. [17], radially integrated to obtain an equivalent stochastic line source.

Grahic Jump Location
Fig. 18

Two-point coherence of reconstructions of the model source CSD: (a) mode 1, (b) modes 1–2, (c) modes 1–3, and (d) all modes (full CSD)

Grahic Jump Location
Fig. 19

Sound radiation by stochastic model sources, decomposed into POD modes. Results for khL = 5 and Lc = L: (a) M = 0.6 and (b) M = 2.0.

Grahic Jump Location
Fig. 20

Two-point coherence of reconstructions of measured nearfield pressure CSD, from Breakey et al. [14]. White represents zero and black shows unit coherence: (a) mode 1, (b) modes 1–2, (c) modes 1–3, and (d) all modes (full CSD).

Grahic Jump Location
Fig. 21

Local spatial instability branches of the Ginzburg–Landau equation, for ω = 1. Solid lines: A = 0.6; dashed lines: A = 1; and dotted lines: A = 1.25.

Grahic Jump Location
Fig. 22

Global stability spectra of the Ginzburg–Landau Eq. (57). (a) Maximum growth rate ω1,i as a function of A, (b) eigenvalue spectrum for the marginally stable case A = 1.6, () exact eigenvalues of the continuous problem, (°) numerical eigenvalues of the discretized problem, and (c) eigenfunction q1(x) associated with the marginally stable eigenvalue for A = 1.6; () real part, () imaginary part, and () modulus.

Grahic Jump Location
Fig. 23

Resolvent gain values of the Ginzburg–Landau system for ω = 1: () A = 1 and (°) A = 0.6

Grahic Jump Location
Fig. 24

The first four resolvent modes of the forced Ginzburg–Landau system, for ω = 1 and A = 1. Legend: () real part, () imaginary part, and () modulus: (a) forcing mode 1, (b) response mode 1, (c) forcing mode 2, (d) response mode 2, (e) forcing mode 3, (f) response mode 3, (g) forcing mode 4, and (h) response mode 4.

Grahic Jump Location
Fig. 25

Forcing and response CSDs (real part) of the stochastically forced Ginzburg–Landau system, for parameters A = 0.6, ω = 1, kh = 1, and Lc as indicated: (a) forcing CSD, Lc = 0.1, (b) forcing CSD, Lc = 1, (c) forcing CSD, Lc = 10, (d) response CSD, Lc = 0.1, (e) response CSD, Lc = 1, and (f) response CSD, Lc = 10

Grahic Jump Location
Fig. 26

The first four SPOD modes of the stochastically forced Ginzburg–Landau system, for ω = 1 and A = 0.6. Left column: low-coherence case Lc = 0.1, right column: high-coherence case Lc = 10. Legend: () real part, () imaginary part, and () modulus. Dotted black lines represent the modulus of the corresponding resolvent response modes: (a) Lc = 0.1, SPOD mode 1, (b) Lc = 10, SPOD mode 1, (c) Lc = 0.1, SPOD mode 2, (d) Lc = 10, SPOD mode 2, (e) Lc = 0.1, SPOD mode 3, (f) Lc = 10, SPOD mode 3, (g) Lc = 0.1, SPOD mode 4, and (h) Lc = 10, SPOD mode 4.

Grahic Jump Location
Fig. 27

First SPOD mode (a, measured) and first resolvent mode (b, computed) in a turbulent jet at St = 0.4, Ma = 0.4 and Re = 460,000. Results show the real part of axial velocity fluctuations. Made with data from Ref. [51].

Grahic Jump Location
Fig. 28

Acoustic radiation of the coupled Ginzburg–Landau/acoustic problem. The thick black line shows the full sound radiation considering the model forcing Pff from Eq. (61). Lower lines show sound radiation with increasing number of SPOD modes of q considered in the acoustic problem (from 1 to 8 SPOD modes).

Grahic Jump Location
Fig. 29

Reconstruction of the radiated sound from a forcing CSD Pff using resolvent modes of Rcoupled. The thick black line shows the full PSD of pressure, and the remaining lines show a reconstruction with 1, 2, or 3 resolvent modes.

Grahic Jump Location
Fig. 30

Resolvent modes of Rcoupled

Grahic Jump Location
Fig. 31

Optimal and first suboptimal x-momentum forcing modes and corresponding responses for a Mach 0.9 jet at St = 0.56, considering acoustic pressure as the output. Gray lines in subfigures (a) and (b) mark the nozzle, and rectangles in subfigures (c) and (d) highlight the domain considered for the acoustic response. Results from Jeun et al. [49]: (a) optimal forcing, (b) first suboptimal forcing, (c) optimal acoustic response, and (d) first suboptimal acoustic response.

Grahic Jump Location
Fig. 32

Leading resolvent gains for various Mach numbers. Gains for the turbulent jet taken from Jeun et al. [49]: (a) coupled Ginzburg–Landau acoustic problem and (b) turbulent jet.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In