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Transition to chaos in a reduced-order model of a shear layer

Published online by Cambridge University Press:  14 December 2021

André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Erico L. Rempel Open the ORCID record for Erico L. Rempel [Opens in a new window]  and
Petrônio A.S. Nogueira Open the ORCID record for Petrônio A.S. Nogueira [Opens in a new window]
André V.G. Cavalieri*
Affiliation:
Divisão de Engenharia Aeroespacial, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil
Erico L. Rempel
Affiliation:
Divisão de Ciências Fundamentais, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brazil
Petrônio A.S. Nogueira
Affiliation:
Department of Mechanical and Aerospace Engineering, Laboratory for Turbulence Research in Aerospace and Combustion, Monash University, Clayton, Australia
*
†Email address for correspondence: andre@ita.br
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Abstract

The present work studies the nonlinear dynamics of a shear layer, driven by a body force and confined between parallel walls, a simplified setting to study transitional and turbulent shear layers. It was introduced by Nogueira & Cavalieri (J. Fluid Mech., vol. 907, 2021, A32), and is here studied using a reduced-order model based on a Galerkin projection of the Navier–Stokes system. By considering a confined shear layer with free-slip boundary conditions on the walls, periodic boundary conditions in streamwise and spanwise directions may be used, simplifying the system and enabling the use of methods of dynamical systems theory. A basis of eight modes is used in the Galerkin projection, representing the mean flow, Kelvin–Helmholtz vortices, rolls, streaks and oblique waves, structures observed in the cited work, and also present in shear layers and jets. A dynamical system is obtained, and its transition to chaos is studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf bifurcations, and the latter leads to a limit cycle with amplitude modulation of vortices, as in the direct numerical simulations by Nogueira & Cavalieri. Further increase of $Re$ leads to the appearance of a chaotic saddle, followed by the emergence of quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging crisis for higher $Re$, leading to a chaotic dynamics with amplitude modulation and phase jumps of vortices. This is reminiscent of observations of coherent structures in turbulent jets, suggesting that the model represents a dynamics consistent with features of shear layers and jets.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Chaos Acoustics: Jet noise
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 932 , 10 February 2022 , A43
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abreu, L.I., Cavalieri, A.V., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020 Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows. J. Fluid Mech. 900, A11.CrossRefGoogle Scholar
Akamine, M., Okamoto, K., Teramoto, S. & Tsutsumi, S. 2019 Conditional sampling analysis of high-speed schlieren movies of mach wave radiation in a supersonic jet. J. Acoust. Soc. Am. 145 (1), EL122.CrossRefGoogle Scholar
Baqui, Y.B., Agarwal, A., Cavalieri, A.V. & Sinayoko, S. 2015 A coherence-matched linear source mechanism for subsonic jet noise. J. Fluid Mech. 776, 235–267.CrossRefGoogle Scholar
Battelino, P.M., Grebogi, C., Ott, E., Yorke, J.A. & Yorke, E.D. 1988 Multiple coexisting attractors, basin boundaries and basic sets. Phys. D: Nonlinear Phenom. 32 (2), 296–305.CrossRefGoogle Scholar
Bogey, C. 2019 On noise generation in low reynolds number temporal round jets at a mach number of 0.9. J. Fluid Mech. 859, 1022–1056.CrossRefGoogle Scholar
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, 507–538.CrossRefGoogle Scholar
Brandt, L. 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. (B/Fluids) 47, 80–96.CrossRefGoogle Scholar
Brès, G.A., Jordan, P., Jaunet, V., Le Rallic, M., Cavalieri, A.V., 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
Brown, G.L. & Roshko, A. 1974 On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64 (4), 775–816.CrossRefGoogle Scholar
Cavalieri, A.V.G. 2021 Structure interactions in a reduced-order model for wall-bounded turbulence. Phys. Rev. Fluids 6 (3), 034610.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. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
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, 559–592.CrossRefGoogle Scholar
Chandler, G.J. & Kerswell, R.R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L.S. & Barkley, D. 2016 Turbulent–laminar patterns in shear flows without walls. J. Fluid Mech. 791, R8.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L.S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.CrossRefGoogle Scholar
Chian, A.C. -L., Borotto, F.A., Rempel, E.L. & Rogers, C. 2005 Attractor merging crisis in chaotic business cycles. Chaos, Solitons Fractals 24 (3), 869–875.CrossRefGoogle Scholar
Cohen, J. & 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, 191–219.CrossRefGoogle Scholar
Crighton, D.G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397–413.CrossRefGoogle Scholar
Crow, S.C. & Champagne, F.H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547–591.CrossRefGoogle Scholar
Cvitanović, P. & Eckhardt, B. 1989 Periodic-orbit quantization of chaotic systems. Phys. Rev. Lett. 63 (8), 823.CrossRefGoogle ScholarPubMed
Del Alamo, J.C. & Jimenez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Duguet, Y., Monokrousos, A., Brandt, L. & Henningson, D.S. 2013 Minimal transition thresholds in plane Couette flow. Phys. Fluids 25 (8), 084103.CrossRefGoogle Scholar
Eckhardt, B. & Mersmann, A. 1999 Transition to turbulence in a shear flow. Phys. Rev. E 60 (1), 509–517.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids (1958–1988) 18 (4), 487–488.CrossRefGoogle Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22 (7), 071704.CrossRefGoogle Scholar
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, 27–51.CrossRefGoogle Scholar
Garnaud, X., Lesshafft, L., Schmid, P. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.CrossRefGoogle Scholar
Gibson, J.F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Grebogi, C., Ott, E., Romeiras, F. & Yorke, J.A. 1987 Critical exponents for crisis-induced intermittency. Phys. Rev. A 36 (11), 5365.CrossRefGoogle ScholarPubMed
Grebogi, C., Ott, E. & Yorke, J.A. 1983 Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation. Phys. Rev. Lett. 50 (13), 935.CrossRefGoogle 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
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287 (1), 317–348.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
Hof, B., Westerweel, J., Schneider, T.M. & Eckhardt, B. 2006 Finite lifetime of turbulence in shear flows. Nature 443 (7107), 59–62.CrossRefGoogle ScholarPubMed
Hsu, G.-H. , Ott, E. & Grebogi, C. 1988 Strange saddles and the dimensions of their invariant manifolds. Phys. Lett. A 127 (4), 199–204.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
Jaunet, V., Jordan, P. & Cavalieri, A. 2017 Two-point coherence of wave packets in turbulent jets. Phys. Rev. Fluids 2 (2), 024604.CrossRefGoogle 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
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195.CrossRefGoogle Scholar
Kaplan, O., Jordan, P., Cavalieri, A.V. & Brès, G.A. 2021 Nozzle dynamics and wavepackets in turbulent jets. J. Fluid Mech. 923, A22.CrossRefGoogle Scholar
Kashinath, K., Waugh, I. & Juniper, M. 2014 Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos. J. Fluid Mech. 761, 399–430.CrossRefGoogle Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Lajús, F., Sinha, A., Cavalieri, A.V.G., Deschamps, C. & Colonius, T. 2019 Spatial stability analysis of subsonic corrugated jets. J. Fluid Mech. 876, 766–791.CrossRefGoogle Scholar
Lesshafft, L., Semeraro, O., Jaunet, V., Cavalieri, A.V.G. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4 (6), 063901.CrossRefGoogle Scholar
Lucas, D. & Kerswell, R.R. 2015 Recurrent flow analysis in spatiotemporally chaotic 2-dimensional kolmogorov flow. Phys. Fluids 27 (4), 045106.CrossRefGoogle Scholar
Marant, M. & Cossu, C. 2018 Influence of optimally amplified streamwise streaks on the Kelvin–Helmholtz instability. J. Fluid Mech. 838, 478–500.CrossRefGoogle Scholar
Matsubara, M., Alfredsson, P.H. & Segalini, A. 2020 Linear modes in a planar turbulent jet. J. Fluid Mech. 888, A26.CrossRefGoogle Scholar
McKeon, B. & Sharma, A. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6 (1), 56.CrossRefGoogle Scholar
Moore, C.J. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80 (2), 321–367.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Nogueira, P.A. & Cavalieri, A.V. 2021 Dynamics of shear-layer coherent structures in a forced wall-bounded flow. J. Fluid Mech. 907, A32.CrossRefGoogle Scholar
Nogueira, P.A., Cavalieri, A.V., Jordan, P. & Jaunet, V. 2019 Large-scale streaky structures in turbulent jets. J. Fluid Mech. 873, 211–237.CrossRefGoogle Scholar
Nogueira, P.A. & Edgington-Mitchell, D.M. 2021 Investigation of supersonic twin-jet coupling using spatial linear stability analysis. J. Fluid Mech. 918, A38.CrossRefGoogle Scholar
Nusse, H.E. & Yorke, J.A. 1989 A procedure for finding numerical trajectories on chaotic saddles. Phys. D: Nonlinear Phenom. 36 (1-2), 137–156.CrossRefGoogle Scholar
Parker, T.S. & Chua, L. 2012 Practical Numerical Algorithms for Chaotic Systems. Springer Science & Business Media.Google Scholar
Pickering, E., Rigas, G., Nogueira, P.A., Cavalieri, A.V., Schmidt, O.T. & Colonius, T. 2020 Lift-up, Kelvin–Helmholtz and Orr mechanisms in turbulent jets. J. Fluid Mech. 896, A2.CrossRefGoogle Scholar
Pierrehumbert, R. & Widnall, S. 1982 The two-and three-dimensional instabilities of a spatially periodic shear layer. J. Fluid Mech. 114, 59–82.CrossRefGoogle Scholar
Rempel, E., Chian, A. -L., Koga, D., Miranda, R. & Santana, W. 2008 Alfvén complexity. Intl J. Bifurcation Chaos 18 (06), 1697–1703.CrossRefGoogle Scholar
Rempel, E.L., Chian, A.C. -L., Macau, E.E. & Rosa, R.R. 2004 Analysis of chaotic saddles in low-dimensional dynamical systems: the derivative nonlinear schrödinger equation. Phys. D: Nonlinear Phenom. 199 (3-4), 407–424.CrossRefGoogle Scholar
Robert, C., Alligood, K.T., Ott, E. & Yorke, J.A. 2000 Explosions of chaotic sets. Phys. D: Nonlinear Phenom. 144 (1-2), 44–61.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., Morfey, C.L. & Hu, Z.W. 2006 Sound radiation from exponentially growing and decaying surface waves. J. Sound Vib. 294 (1), 355–361.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows, vol. 142. Springer.CrossRefGoogle Scholar
Schmidt, O.T. 2020 Bispectral mode decomposition of nonlinear flows. Nonlinear Dyn. 102 (4), 2479–2501.CrossRefGoogle Scholar
Schmidt, O.T. & Schmid, P.J. 2019 A conditional space–time pod formalism for intermittent and rare events: example of acoustic bursts in turbulent jets. J. Fluid Mech. 867, R2.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
Schneider, T.M., Eckhardt, B. & Yorke, J.A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99 (3), 034502.CrossRefGoogle ScholarPubMed
Skufca, J.D., Yorke, J.A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96 (17), 174101.CrossRefGoogle Scholar
Suponitsky, V., Sandham, N. & Morfey, C. 2010 Linear and nonlinear mechanisms of sound radiation by instability waves in subsonic jets. J. Fluid Mech. 658, 509–538.CrossRefGoogle Scholar
Suzuki, T. & Colonius, T. 2006 Instability waves in a subsonic round jet detected using a near-field phased microphone array. J. Fluid Mech. 565, 197–226.CrossRefGoogle Scholar
Tam, C.K.W. & Burton, D.E. 1984 Sound generated by instability waves of supersonic flows. Part 2. Axisymmetric jets. J. Fluid Mech. 138 (138), 273–295.CrossRefGoogle Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–316.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Waleffe, F. 1995 Transition in shear flows. Nonlinear normality versus non-normal linearity. Phys. Fluids 7 (12), 3060–3066.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Wang, C., Lesshafft, L., Cavalieri, A.V. & Jordan, P. 2021 The effect of streaks on the instability of jets. J. Fluid Mech. 910, A14.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Wu, X. & Huerre, P. 2009 Low-frequency sound radiated by a nonlinearly modulated wavepacket of helical modes on a subsonic circular jet. J. Fluid Mech. 637, 173–211.CrossRefGoogle Scholar
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. & Wu, X. 2020 Nonlinear evolution and acoustic radiation of coherent structures in subsonic turbulent free shear layers. J. Fluid Mech. 884, A10.CrossRefGoogle Scholar