Skip to main content Accessibility help
×
Home

Spectral analysis of jet turbulence

  • Oliver T. Schmidt (a1), Aaron Towne (a2), Georgios Rigas (a1), Tim Colonius (a1) and Guillaume A. Brès (a3)...

Abstract

Informed by large-eddy simulation (LES) data and resolvent analysis of the mean flow, we examine the structure of turbulence in jets in the subsonic, transonic and supersonic regimes. Spectral (frequency-space) proper orthogonal decomposition is used to extract energy spectra and decompose the flow into energy-ranked coherent structures. The educed structures are generally well predicted by the resolvent analysis. Over a range of low frequencies and the first few azimuthal mode numbers, these jets exhibit a low-rank response characterized by Kelvin–Helmholtz (KH) type wavepackets associated with the annular shear layer up to the end of the potential core and that are excited by forcing in the very-near-nozzle shear layer. These modes too have been experimentally observed before and predicted by quasi-parallel stability theory and other approximations – they comprise a considerable portion of the total turbulent energy. At still lower frequencies, particularly for the axisymmetric mode, and again at high frequencies for all azimuthal wavenumbers, the response is not low-rank, but consists of a family of similarly amplified modes. These modes, which are primarily active downstream of the potential core, are associated with the Orr mechanism. They occur also as subdominant modes in the range of frequencies dominated by the KH response. Our global analysis helps tie together previous observations based on local spatial stability theory, and explains why quasi-parallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.

Copyright

Corresponding author

Email address for correspondence: oschmidt@ucsd.edu

Footnotes

Hide All

Present address: Department of Mechanical and Aerospace Engineering, University of California at San Diego, La Jolla, CA 92093, USA.

§

Present address: Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA.

Footnotes

References

Hide All
Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.
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, 133.
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. J. Theor. Comput. Fluid Dyn. 2 (5), 339352.
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485504.
Bishop, K. A., Ffowcs Williams, J. E. & Smith, W. 1971 On the noise sources of the unsuppressed high-speed jet. J. Fluid Mech. 50 (1), 2131.
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.
Brès, G. A., Jordan, P., Jaunet, V., Le Rallic, M., 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, 83124.
Cavalieri, A. V. G., Jordan, P., Agarwal, A. & Gervais, Y. 2011 Jittering wave-packet models for subsonic jet noise. J. Sound Vib. 330 (18), 44744492.
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, 559592.
Cavalieri, A. V. G., Sasaki, K., Jordan, P., Schmidt, O. T., Colonius, T. & Brès, G. A. 2016 High-frequency wavepackets in turbulent jets. In 22nd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2016-3056.
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215234.
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, 137166.
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397413.
Crighton, D. G. & Huerre, P. 1990 Shear-layer pressure fluctuations and superdirective acoustic sources. J. Fluid Mech. 220, 355368.
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547591.
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, 406430.
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5 (11), 26002609.
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189202.
Glauser, M. N., Leib, S. J. & George, W. K. 1987 Coherent structures in the axisymmetric turbulent jet mixing layer. In Turbulent Shear Flows (ed. Durst, F. et al. ), vol. 5, pp. 134145. Springer.
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97128.
Jeun, J., Nichols, J. W. & Jovanović, M. R. 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173195.
Jordan, P., Zhang, M., Lehnasch, G. & Cavalieri, A. V. G. 2017 Modal and non-modal linear wavepacket dynamics in turbulent jets. In 23rd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2017-3379.
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166178. Nauka.
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.
Marquet, O., Lombardi, M., Chomaz, J.-M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 121.
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.
Mettot, C., Sipp, D. & Bézard, H. 2014 Quasi-laminar stability and sensitivity analyses for turbulent flows: prediction of low-frequency unsteadiness and passive control. Phys. Fluids 26 (4), 045112.
Michalke, A. 1971 Instability of a compressible circular free jet with consideration of the influence of the jet boundary layer thickness. Z. Flugwiss. 19 (8), 319328.
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205240.
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181214.
Moore, C. J. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80 (2), 321367.
Pope, S. B. 2000 Turbulent Flows, 1st edn. Cambridge University Press.
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252 (1), 209238.
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 1547.
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.
Schlinker, R. H., Simonich, J. C., Shannon, D. W., Reba, R. A., Colonius, T., Gudmundsson, K. & Ladeinde, F. 2009 Supersonic jet noise from round and chevron nozzles: experimental studies. AIAA Paper 2009-3257.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
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, 11531181.
Semeraro, O., Jaunet, V., Jordan, P., Cavalieri, A. V. G. & Lesshafft, L.2016a Stochastic and harmonic optimal forcing in subsonic jets. In 22nd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2016-2935.
Semeraro, O., Lesshafft, L., Jaunet, V. & Jordan, P. 2016b Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: theory and experiment. Intl J. Heat Fluid Flow 62, 2432.
Sinha, A., Rodríguez, D., Brès, G. A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 7195.
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617635.
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. Appl. Maths 45 (3), 561571.
Suzuki, T. & Colonius, T. 2006 Instability waves in a subsonic round jet detected using a near-field phased microphone array. J. Fluid Mech. 565 (1), 197226.
Tam, C. K. W. & Hu, F. Q. 1989 On the three families of instability waves of high-speed jets. J. Fluid Mech. 201, 447483.
Tissot, G., Zhang, M., Lajús, F. C., Cavalieri, A. V. G. & Jordan, P. 2017 Sensitivity of wavepackets in jets to nonlinear effects: the role of the critical layer. J. Fluid Mech. 811, 95137.
Towne, A., Brès, G. A. & Lele, S. K.2017a A statistical jet-noise model based on the resolvent framework. In 23rd AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2017-3706.
Towne, A., Cavalieri, A. V. G., Jordan, P., Colonius, T., Schmidt, O. T., Jaunet, V. & Brès, G. A. 2017b Acoustic resonance in the potential core of subsonic jets. J. Fluid Mech. 825, 11131152.
Towne, A., Colonius, T., Jordan, P., Cavalieri, A. V. G. & Brès, G. A. 2015 Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet. In 21st AIAA/CEAS Aeroacoustics Conference. AIAA Paper 2015-2217.
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.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578584.
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636680.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Related content

Powered by UNSILO

Spectral analysis of jet turbulence

  • Oliver T. Schmidt (a1), Aaron Towne (a2), Georgios Rigas (a1), Tim Colonius (a1) and Guillaume A. Brès (a3)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.