Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T22:06:21.512Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral analysis of jet turbulence

Published online by Cambridge University Press:  21 September 2018

Oliver T. Schmidt Open the ORCID record for Oliver T. Schmidt [Opens in a new window] ,
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window] ,
Georgios Rigas Open the ORCID record for Georgios Rigas [Opens in a new window] ,
Tim Colonius  and
Guillaume A. Brès Open the ORCID record for Guillaume A. Brès [Opens in a new window]
Oliver T. Schmidt*
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Aaron Towne
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
Georgios Rigas
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Tim Colonius
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
Guillaume A. Brès
Affiliation:
Cascade Technologies Inc., Palo Alto, CA 94303, USA
*
†Email address for correspondence: oschmidt@ucsd.edu
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Instability: Absolute/convective instability Acoustics: Jet noise Turbulent Flows: Shear layer turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 953 - 982
Check for updates
Copyright
© 2018 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.)

Footnotes

‡

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.

References

Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.Google Scholar
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, 1–33.Google Scholar
Aubry, N. 1991 On the hidden beauty of the proper orthogonal decomposition. J. Theor. Comput. Fluid Dyn. 2 (5), 339–352.Google 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.Google Scholar
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), 21–31.Google 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), 1164–1184.Google Scholar
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, 83–124.Google 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), 4474–4492.Google 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.Google Scholar
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.Google Scholar
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215–234.Google Scholar
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, 137–166.Google Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397–413.Google Scholar
Crighton, D. G. & Huerre, P. 1990 Shear-layer pressure fluctuations and superdirective acoustic sources. J. Fluid Mech. 220, 355–368.Google Scholar
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547–591.Google 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.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5 (11), 2600–2609.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.Google Scholar
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. 134–145. Springer.Google Scholar
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128.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.Google Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195.Google Scholar
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.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.Google Scholar
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Propagation (ed. Yaglom, A. M. & Tatarski, V. I.), pp. 166–178. Nauka.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
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, 1–21.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
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.Google Scholar
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), 319–328.Google Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.Google Scholar
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, 181–214.Google Scholar
Moore, C. J. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80 (2), 321–367.Google Scholar
Pope, S. B. 2000 Turbulent Flows, 1st edn. Cambridge University Press.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252 (1), 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 15–47.Google Scholar
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.Google Scholar
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.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, 1st edn. Springer.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
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, 1153–1181.Google Scholar
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.Google Scholar
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, 24–32.Google Scholar
Sinha, A., Rodríguez, D., Brès, G. A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 71–95.Google Scholar
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), 617–635.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. Appl. Maths 45 (3), 561–571.Google 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 (1), 197–226.Google Scholar
Tam, C. K. W. & Hu, F. Q. 1989 On the three families of instability waves of high-speed jets. J. Fluid Mech. 201, 447–483.Google Scholar
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, 95–137.Google Scholar
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.Google Scholar
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, 1113–1152.Google Scholar
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.Google 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, 821–867.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.Google Scholar