Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-09T21:55:15.508Z Has data issue: false hasContentIssue false
  • English
  • Français

Sensitivity of wavepackets in jets to nonlinear effects: the role of the critical layer

Published online by Cambridge University Press:  06 December 2016

Gilles Tissot ,
Mengqi Zhang ,
Francisco C. Lajús Jr ,
André V. G. Cavalieri  and
Peter Jordan
Gilles Tissot*
Affiliation:
Instituto Tecnológico de Aeronáutica, Divisão de Engenharia Aeronáutica, São José dos Campos, 12228-900 SP, Brazil
Mengqi Zhang
Affiliation:
Institut PPRIME, Fluid, Thermal and Combustion Department, 8692 Poitiers, France
Francisco C. Lajús Jr
Affiliation:
Instituto Tecnológico de Aeronáutica, Divisão de Engenharia Aeronáutica, São José dos Campos, 12228-900 SP, Brazil Universidade Federal de Santa Catarina, Depart. de Eng. Mecânica, Florianópolis, 88040-900 SC, Brazil
André V. G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, Divisão de Engenharia Aeronáutica, São José dos Campos, 12228-900 SP, Brazil
Peter Jordan
Affiliation:
Institut PPRIME, Fluid, Thermal and Combustion Department, 8692 Poitiers, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear instability waves, or wavepackets, are key building blocks for the jet-noise problem. It has been shown in previous work that linear models correctly predict the evolution of axisymmetric wavepackets up to the end of the potential core of subsonic turbulent jets. Beyond this station, linear models fail, and nonlinearity is the likely missing piece. The essential underlying nonlinear mechanisms are unknown, and it remains unclear how these should be incorporated in a reduced-order model. The nonlinear interactions are considered in this work as an ‘external’ harmonic forcing added to the standard linear model. This modelling framework is explored using a locally parallel resolvent analysis to determine optimal forcing and associated responses, and a global approach based on 4D-Var data assimilation aimed at finding the optimal forcing of the parabolised stability equations that would minimise errors in the predictions of wavepackets. In all of the problems considered, the critical layer is found to be relevant: it is the position where sensitivity of wavepackets to nonlinearity is greatest. It is seen that disturbances are forced around the critical layer, and tilted by shear as they are advected, in a manner suggestive of an Orr-like mechanism. The ensemble of results suggests that critical-layer effects play a central role in the dynamics of wavepackets in subsonic turbulent jets, and that inclusion of such effects may remedy the shortcomings of linear reduced-order models.

JFM classification

Instability: Absolute/convective instability Acoustics: Jet noise
Type
Papers
Information
Journal of Fluid Mechanics , Volume 811 , 25 January 2017 , pp. 95 - 137
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adam, J. 1984 The critical layers and other singular regions in ideal hydrodynamics and magnetohydrodynamics. Astrophys. Space Sci. 105 (2), 401–412.CrossRefGoogle Scholar
Airiau, C., Bottaro, A., Walther, S. & Legendre, D. 2003 A methodology for optimal laminar flow control: application to the damping of Tollmien–Schlichting waves in a boundary layer. Phys. Fluids 15 (5), 1131–1145.CrossRefGoogle Scholar
Alazard, T. 2006 Low Mach number limit of the full Navier–Stokes equations. Arch. Rat. Mech. Anal. 180 (1), 1–73.CrossRefGoogle Scholar
Ansaldi, A. & Airiau, C.2015 Sensitivity analysis for subsonic jet using adjoint of non local stability equations. In 21th AIAA/CEAS Aeroacoustic Conference and Exhibit. 22–26 June, Dallas, Texas, AIAA Paper 2015-2219.Google Scholar
Baqui, Y. B., Agarwal, A., Cavalieri, A. V. G. & Sinayoko, S. 2015 A coherence-matched linear source mechanism for subsonic jet noise. J. Fluid Mech. 776, 235–267.CrossRefGoogle Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504.CrossRefGoogle Scholar
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179–225.CrossRefGoogle Scholar
Brambley, E. J., Darau, M. & Rienstra, S. W. 2012 The critical layer in linear-shear boundary layers over acoustic linings. J. Fluid Mech. 710, 545–568.CrossRefGoogle Scholar
Breakey, D. E. S., Jordan, P., Cavalieri, A. V. G., Léon, O., Zhang, M., Lehnasch, G., Colonius, T. & Rodriguez, D.2013 Near-field wavepackets and the far-field sound of a subsonic jet. In 19th AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2013-2083.Google Scholar
Campos, L. M. B. C., Oliveira, J. M. G. S. & Kobayashi, M. H. 1999 On sound propagation in a linear shear flow. J. Sound Vib. 219 (5), 739–770.CrossRefGoogle Scholar
Cavalieri, A. V. G. & Agarwal, A. 2014 Coherence decay and its impact on sound radiation by wavepackets. J. Fluid Mech. 748, 399–415.CrossRefGoogle Scholar
Cavalieri, A. V. G., Daviller, G., Comte, P., Jordan, P., Tadmor, G. & Gervais, Y. 2011a Using large eddy simulation to explore sound-source mechanisms in jets. J. Sound Vib. 330 (17), 4098–4113.CrossRefGoogle Scholar
Cavalieri, A. V. G., Jordan, P., Agarwal, A. & Gervais, Y. 2011b Jittering wave-packet models for subsonic jet noise. J. Sound Vib. 330 (18), 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., 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
Cordier, L., El Majd, B. A. & Favier, J. 2010 Calibration of POD reduced-order models using Tikhonov regularization. Intl J. Numer. Meth. Fluids 63 (2), 269–296.CrossRefGoogle Scholar
Cordier, L., Noack, B. R., Daviller, G., Tissot, G., Lehnasch, G., Delville, J., Balajewicz, M. & Niven, R. 2013 Identification strategies for model-based control. Exp. Fluids 54 (8), 1–21.CrossRefGoogle Scholar
Cowley, S. J. & Wu, X.-S. 1994 Asymptotic approaches to transition modelling. In AGARD, Special Course on Progress in Transition Modelling 38 p (SEE N94-33884 10-34), vol. 1.Google Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397–413.CrossRefGoogle Scholar
Dergham, G., Sipp, D. & Robinet, J.-C. 2013 Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow. J. Fluid Mech. 719, 406–430.CrossRefGoogle Scholar
Dobrinsky, A.2002 Adjoint analysis for receptivity prediction. PhD thesis, Rice University, Houston, Texas.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Ffowcs-Williams, J. E. & Kempton, A. J. 1978 The noise from the large-scale structure of a jet. J. Fluid Mech. 84 (04), 673–694.CrossRefGoogle Scholar
Freund, J. B.1997 Compressibility effects in a turbulent annular mixing layer. PhD thesis, Stanford University.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.CrossRefGoogle Scholar
Gloor, M., Obrist, D. & Kleiser, L. 2013 Linear stability and acoustic characteristics of compressible, viscous, subsonic coaxial jet flow. Phys. Fluids 25 (8), 084102.CrossRefGoogle Scholar
Gudmundsson, K.2010 Instability wave models of turbulent jets from round and serrated nozzles. PhD thesis, California Institute of Technology, Pasadena, California.Google Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128.CrossRefGoogle Scholar
Gunzburger, M. D. 2003 Perspectives in Flow Control and Optimization. SIAM.Google Scholar
Haberman, R. 1976 Nonlinear perturbations of the Orr–Sommerfeld equation – asymptotic expansion of the logarithmic phase shift across the critical layer. SIAM J. Math. Anal. 7 (1), 70–81.CrossRefGoogle Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–837.CrossRefGoogle Scholar
Hansen, P. C. 1992 Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34 (4), 561–580.CrossRefGoogle Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.CrossRefGoogle Scholar
Huerre, P. 1980 The nonlinear stability of a free shear layer in the viscous critical layer regime. Phil. Trans. R. Soc. Lond. A 293 (1408), 643–672.Google Scholar
Huerre, P. & Scott, J. F. 1980 Effects of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371 (1747), 509–524.Google Scholar
Jeun, J., Nichols, J. W. & Jovanović, M. R. 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.CrossRefGoogle Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195.CrossRefGoogle Scholar
Jordan, P., Colonius, T., Brès, G. A., Zhang, M., Towne, A. & Lele, S. K. 2014 Modeling intermittent wavepackets and their radiated sound in a turbulent jet. In Proceedings of the Summer Program, Center for Turbulence Research, Stanford University.Google Scholar
Jordan, P. & Gervais, Y. 2008 Subsonic jet aeroacoustics: associating experiment, modelling and simulation. Exp. Fluids 44 (1), 1–21.CrossRefGoogle Scholar
Kerhervé, F., Jordan, P., Cavalieri, A. V. G., Delville, J., Bogey, C. & Juvé, D. 2012 Educing the source mechanism associated with downstream radiation in subsonic jets. J. Fluid Mech. 710, 606–640.CrossRefGoogle Scholar
Landahl, M. T. 1967 A wave-guide model for turbulent shear flow. J. Fluid Mech. 29, 441–459.CrossRefGoogle Scholar
Lesshafft, L. 2015 Preface to this Festschrift for Patrick Huerre. Eur. J. Mech. (B/Fluids) 49, 299–300.CrossRefGoogle Scholar
Lin, C. C. 1954 Some physical aspects of the stability of parallel flows. Proc. Natl Acad. Sci. USA 40 (8), 741–747.CrossRefGoogle ScholarPubMed
Maslowe, S. A. 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18 (1), 405–432.CrossRefGoogle Scholar
Mckeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Meseguer, Á. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186 (1), 178–197.CrossRefGoogle Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199.CrossRefGoogle Scholar
Moarref, R., Jovanović, M. R., Tropp, J. A., Sharma, A. S. & Mckeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26 (5), 051701.CrossRefGoogle Scholar
Navon, I. M. 2009 Data assimilation for numerical weather prediction: a review. In Data Assimilation for Atmospheric, Oceanic, and Hydrologic Applications. Springer.Google Scholar
Nichols, J. & Jovanović, M. 2014 Input–ouput analysis of high-speed jet noise. In Proceedings of the Summer Program, Center for Turbulence Research, Stanford University.Google Scholar
Nocedal, J. & Wright, St. J. 1999 Numerical Optimization. Springer.CrossRefGoogle Scholar
Papadakis, N.2007 Assimilation de données images: application au suivi de courbes et de champs de vecteurs. PhD thesis, Université de Rennes I.Google Scholar
Poinsot, T. J. & Lele, S. K. 1992 Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101 (1), 104–129.CrossRefGoogle Scholar
Pralits, J. O., Airiau, C., Hanifi, A. & Henningson, D. S. 2000 Sensitivity analysis using adjoint parabolized stability equations for compressible flows. Flow Turbul. Combust. 65 (3–4), 321–346.CrossRefGoogle Scholar
Pralits, J. O., Hanifi, A. & Henningson, D. S. 2002 Adjoint-based optimization of steady suction for disturbance control in incompressible flows. J. Fluid Mech. 467, 129–161.CrossRefGoogle Scholar
Rowley, C. W., Colonius, T. & Murray, R. M. 2004 Model reduction for compressible flows using POD and Galerkin projection. Physica D 189 (1–2), 115–129.CrossRefGoogle Scholar
Sasaki, K.2015 Estudo e controle de pacotes de onda em jatos utilizando as equaões de estabilidade parabolizadas. Master thesis, Instituto Technológico de Aeronáutica, São José dos Campos, Brazil.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. vol. 142. Springer.CrossRefGoogle Scholar
Semeraro, O., Jaunet, V., Jordan, P., Cavalieri, A. V. G. & Lesshafft, L.2016 Stochastic and harmonic optimal forcing in subsonic jets. In 22nd AIAA/CEAS Aeroacoustics Conference, AIAA Paper 2016-2935.Google Scholar
Sharma, A. S. & Mckeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Tissot, G., Zhang, M., Lajús, F. C. Jr., Cavalieri, A. V. G., Jordan, P. & Colonius, T. 2015 Sensitivity of wavepackets in jets to non-linear effects: the role of the critical layer. In 21th AIAA/CEAS Aeroacoustic Conference and Exhibit. 22–26 June, Dallas, Texas, AIAA Paper 2015-2218.Google Scholar
Weideman, J. A. & Reddy, S. C. 2000 A Matlab differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.CrossRefGoogle Scholar
Zhang, M., Jordan, P., Lehnasch, G., Cavalieri, A. V. G. & Agarwal, A.2014 Just enough jitter for jet noise? In 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, AIAA Paper 2014-3061.Google Scholar