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Nozzle dynamics and wavepackets in turbulent jets

Published online by Cambridge University Press:  28 July 2021

Oğuzhan Kaplan*
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS – Université de Poitiers – ENSMA, 86036 Poitiers, France Univ. Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des fluides de Lille – Kampé de Fériet, F-59000 Lille, France
Peter Jordan
Affiliation:
Département Fluides, Thermique et Combustion, Institut Pprime, CNRS – Université de Poitiers – ENSMA, 86036 Poitiers, France
André V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeronautica, Instituto Tecnologico de Aeronautica, 12228-900São Jose dos Campos, SP, Brazil
Guillaume A. Brès
Affiliation:
Cascade Technologies Inc., Palo Alto, CA94303, USA
*
Email address for correspondence: oguzhankpln@gmail.com

Abstract

We study a turbulent jet issuing from a cylindrical nozzle to characterise coherent structures evolving in the turbulent boundary layer. The analysis is performed using data from a large-eddy simulation of a Mach 0.4 jet. Azimuthal decomposition of the velocity field in the nozzle shows that turbulent kinetic energy predominantly resides in high azimuthal wavenumbers; the first three azimuthal wavenumbers, that are important for sound generation, contain much lower, but non-zero amplitudes. Using two-point statistics, low azimuthal modes in the nozzle boundary layer are shown to exhibit significant correlations with modes of the same order in the free-jet region. Spectral proper orthogonal decomposition is used to distill a low-rank approximation of the flow dynamics. This reveals the existence of tilted coherent structures within the nozzle boundary layer and shows that these are coupled with wavepackets in the jet. The educed nozzle boundary-layer structures are modelled using a global resolvent analysis of the mean flow inside the nozzle to determine the most amplified flow responses using the linearised Navier–Stokes system. It is shown that the most-energetic nozzle structures can be successfully described with optimal resolvent response modes, whose associated forcing modes are observed to tilt against the nozzle boundary layer, suggesting that the Orr mechanism underpins these organised, turbulent, boundary-layer structures.

Type
JFM Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abreu, L.I., Cavalieri, A.V.G., 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
Batt, R.G. 1975 Layer some measurements on the effect of tripping the two-dimensional shear. AIAA J. 13 (2), 245247.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, 485504.CrossRefGoogle Scholar
Böberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43 (8–9), 697726.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, 507538.CrossRefGoogle Scholar
Bradshaw, P. 1966 The effect of initial conditions on the development of a free shear layer. J. Fluid Mech. 26 (2), 225236.CrossRefGoogle Scholar
Breakey, D.E., Jordan, P., Cavalieri, A.V., Nogueira, P.A., Léon, O., Colonius, T. & Rodríguez, D. 2017 Experimental study of turbulent-jet wave packets and their acoustic efficiency. Phys. Rev. Fluids 2 (12), 124601.CrossRefGoogle Scholar
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.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, 83124.CrossRefGoogle Scholar
Brès, G.A. & Lele, S.K. 2019 Modelling of jet noise: a perspective from large-eddy simulations. Phil. Trans. R. Soc. Lond. A 377 (2159), 20190081.Google ScholarPubMed
Bridges, J. & Hussain, A. 1987 Roles of initial condition and vortex pairing in jet noise. J. Sound Vib. 117 (2), 289311.CrossRefGoogle Scholar
Cavalieri, A.V., 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., Rodríguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559592.CrossRefGoogle Scholar
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215234.CrossRefGoogle Scholar
Citriniti, J. & George, W.K. 2000 Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J. Fluid Mech. 418, 137166.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large–scale structures in turbulent boundary layers. J. Fluid Mech. 619, 7994.CrossRefGoogle Scholar
Crighton, D. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397413.CrossRefGoogle Scholar
Crow, S.C. & Champagne, F. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547591.CrossRefGoogle Scholar
Eitel-Amor, G., Örlü, R. & Schlatter, P. 2014 Simulation and validation of a spatially evolving turbulent boundary layer up to $Re\theta = 8300$. Intl J. Heat Fluid Flow 47, 5769.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, 2751.CrossRefGoogle Scholar
Freund, J. & Colonius, T. 2009 Turbulence and sound-field pod analysis of a turbulent jet. Intl J. Aeroacoust. 8 (4), 337354.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, 189202.CrossRefGoogle Scholar
Gennaro, E., Rodríguez, D., Medeiros, M. & Theofilis, V. 2013 Sparse techniques in global flow instability with application to compressible leading-edge flow. AIAA J. 51 (9), 22952303.CrossRefGoogle Scholar
Grosch, C.E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87 (1), 3354.CrossRefGoogle Scholar
Grosche, F. 1974 Distributions of sound source intensities in subsonic and supersonic jets. In AGARD, Conference Proceedings, 1974, vol. 131, pp. 4-1–4-10.Google Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97128.CrossRefGoogle Scholar
Hanifi, A., Schmid, P.J. & Henningson, D.S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826837.CrossRefGoogle Scholar
Hill, W.G. Jr., Jenkins, R.C. & Gilbert, B.L. 1976 Effects of the initial boundary-layer state on turbulent jet mixing. AIAA J. 14 (11), 15131514.CrossRefGoogle Scholar
Hussain, A. & Zedan, M. 1978 a Effects of the initial condition on the axisymmetric free shear layer: effect of the initial fluctuation level. Phys. Fluids 21 (9), 14751481.CrossRefGoogle Scholar
Hussain, A. & Zedan, M. 1978 b Effects of the initial condition on the axisymmetric free shear layer: effects of the initial momentum thickness. Phys. Fluids 21 (7), 11001112.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, 5173.CrossRefGoogle Scholar
Jiménez, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25 (11), 110814.CrossRefGoogle Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173195.CrossRefGoogle Scholar
Jordan, P., Zhang, M., Lehnasch, G. & Cavalieri, A.V. 2017 Modal and non-modal linear wavepacket dynamics in turbulent jets. In 23rd AIAA/CEAS Aeroacoustics Conference, p. 3379.Google Scholar
Kaiser, T.L., Lesshafft, L. & Oberleithner, K. 2019 Prediction of the flow response of a turbulent flame to acoustic pertubations based on mean flow resolvent analysis. Trans. ASME J. Engng Gas Turbines Power 141 (11), 111021.CrossRefGoogle Scholar
Khorrami, M.R., Malik, M.R. & Ash, R.L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81 (1), 206229.CrossRefGoogle Scholar
Kovesi, P. 2015 Good colour maps: how to design them. arXiv:1509.03700.CrossRefGoogle Scholar
Lesshafft, L., Semeraro, O., Jaunet, V., Cavalieri, A.V. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4 (6), 063901.CrossRefGoogle Scholar
Li, F. & Malik, M.R. 1997 Spectral analysis of parabolized stability equations. Comput. Fluids 26 (3), 279297.CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A.M. Yaglom & V.I. Tatarsky). Nauka.Google Scholar
Maestrello, L. & McDaid, E. 1971 Acoustic characteristics of a high-subsonic jet. AIAA J. 9 (6), 10581066.CrossRefGoogle Scholar
Martini, E., Rodríguez, D., Towne, A. & Cavalieri, A. 2021 Efficient computation of global resolvent modes. J. Fluid Mech. 919, A3.CrossRefGoogle Scholar
McKeon, B. & Sharma, A. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.CrossRefGoogle Scholar
Michalke, A. 1971 Instabilitaet eines Kompressiblen Runden Freistrahis unter Beruecksichtigung des Einflusses der Strahigrenzschichtdicke (Instability of a compressible circular jet considering the influence of the thickness of the jet boundary layer). Tech. Rep. Deutsche Forschungs-und Versuchsanstalt für Luft-und Raumfahrt.Google Scholar
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159199.CrossRefGoogle Scholar
Mollo-Christensen, E. 1967 Jet noise and shear flow instability seen from an experimenter's viewpoint. J. Appl. Mech. 34 (1), 17.CrossRefGoogle Scholar
Monkewitz, P.A., Chauhan, K.A. & Nagib, H.M. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19 (11), 115101.CrossRefGoogle Scholar
Moore, C. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80 (2), 321367.CrossRefGoogle Scholar
Morra, P., Nogueira, P.A.S., Cavalieri, A.V.G. & Henningson, D.S. 2021 The colour of forcing statistics in resolvent analyses of turbulent channel flows. J. Fluid Mech. 907, A24.CrossRefGoogle Scholar
Muralidhar, S.D., Podvin, B., Mathelin, L. & Fraigneau, Y. 2019 Spatio-temporal proper orthogonal decomposition of turbulent channel flow. J. Fluid Mech. 864, 614639.CrossRefGoogle Scholar
Nogueira, P.A., Cavalieri, A.V., Jordan, P. & Jaunet, V. 2019 Large-scale streaky structures in turbulent jets. J. Fluid Mech. 873, 211237.CrossRefGoogle Scholar
Nogueira, P.A.S., Morra, P., Martini, E., Cavalieri, A.V.G. & Henningson, D.S. 2021 Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow. J. Fluid Mech. 908, A32.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
Rodríguez, D., Cavalieri, A.V., Colonius, T. & 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, 308321.CrossRefGoogle Scholar
Sano, A., Abreu, L.I., Cavalieri, A.V. & Wolf, W.R. 2019 Trailing-edge noise from the scattering of spanwise-coherent structures. Phys. Rev. Fluids 4 (9), 094602.CrossRefGoogle Scholar
Sasaki, K., Cavalieri, A.V., Jordan, P., Schmidt, O.T., Colonius, T. & Brès, G.A. 2017 High-frequency wavepackets in turbulent jets. J. Fluid Mech. 830, R2.CrossRefGoogle Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116126.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2016 Boundary-Layer Theory. Springer.Google Scholar
Schmidt, O.T., Towne, A., Rigas, G., Colonius, T. & Brès, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953982.CrossRefGoogle Scholar
Semeraro, O., Lesshafft, L., Jaunet, V. & Jordan, P. 2016 Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: theory and experiment. Intl J. Heat Fluid Flow 62, 2432.CrossRefGoogle Scholar
Sharma, A.S. & McKeon, B.J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.CrossRefGoogle Scholar
Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High–Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353375.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, 197226.CrossRefGoogle Scholar
Tissot, G., Lajús, F.C. Jr., Cavalieri, A.V. & Jordan, P. 2017 a Wave packets and Orr mechanism in turbulent jets. Phys. Rev. Fluids 2 (9), 093901.CrossRefGoogle Scholar
Tissot, G., Zhang, M., Lajús, F.C., Cavalieri, A.V. & Jordan, P. 2017 b Sensitivity of wavepackets in jets to nonlinear effects: the role of the critical layer. J. Fluid Mech. 811, 95137.CrossRefGoogle Scholar
Towne, A., Schmidt, O.T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821867.CrossRefGoogle Scholar
Tumin, A. 1996 Receptivity of pipe Poiseuille flow. J. Fluid Mech. 315, 119137.CrossRefGoogle Scholar
Viswanathan, K. & Clark, L. 2004 Effect of nozzle internal contour on jet aeroacoustics. Intl J. Aeroacoust. 3 (2), 103135.CrossRefGoogle Scholar
Zaman, K. 1985 Effect of initial condition on subsonic jet noise. AIAA J. 23 (9), 13701373.CrossRefGoogle Scholar
Zaman, K. 2012 Effect of initial boundary-layer state on subsonic jet noise. AIAA J. 50 (8), 17841795.CrossRefGoogle Scholar
Zaman, K. & Hussain, A. 1981 Turbulence suppression in free shear flows by controlled excitation. J. Fluid Mech. 103, 133159.CrossRefGoogle Scholar

Kaplan at al. Supplementary Movie

Cross-spectral-density of axisymmetric streamwise velocity fluctuations with the correlation point indicated indicated by `+' for St = 0:6.

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