Hostname: page-component-7dd5485656-bt4hw Total loading time: 0 Render date: 2025-10-31T06:13:35.125Z Has data issue: false hasContentIssue false
  • English
  • Français

Ambiguity in mean-flow-based linear analysis

Published online by Cambridge University Press:  06 August 2020

U. Karban Open the ORCID record for U. Karban [Opens in a new window] ,
B. Bugeat ,
E. Martini Open the ORCID record for E. Martini [Opens in a new window] ,
A. Towne Open the ORCID record for A. Towne [Opens in a new window] ,
A. V. G. Cavalieri Open the ORCID record for A. V. G. Cavalieri [Opens in a new window] ,
L. Lesshafft Open the ORCID record for L. Lesshafft [Opens in a new window] ,
A. Agarwal ,
P. Jordan Open the ORCID record for P. Jordan [Opens in a new window]  and
T. Colonius
U. Karban*
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France
B. Bugeat
Affiliation:
Department of Engineering, University of Cambridge, Cambridge CB2 3AP, UK
E. Martini
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
A. Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI48109, USA
A. V. G. Cavalieri
Affiliation:
Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil
L. Lesshafft
Affiliation:
Laboratoire d'Hydrodynamique, CNRS – École Polytechnique – Institut Polytechnique de Paris, Palaiseau, France
A. Agarwal
Affiliation:
Department of Engineering, University of Cambridge, Cambridge CB2 3AP, UK
P. Jordan
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS – University of Poitiers – ENSMA, Poitiers, France
T. Colonius
Affiliation:
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA91125, USA
*
†Email address for correspondence: ugur.karban@univ-poitiers.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Linearisation of the Navier–Stokes equations about the mean of a turbulent flow forms the foundation of popular models for energy amplification and coherent structures, including resolvent analysis. While the Navier–Stokes equations can be equivalently written using many different sets of dependent variables, we show that the properties of the linear operator obtained via linearisation about the mean depend on the variables in which the equations are written prior to linearisation, and can be modified under nonlinear transformation of variables. For example, we show that using primitive and conservative variables leads to differences in the singular values and modes of the resolvent operator for turbulent jets, and that the differences become more severe as variable-density effects increase. This lack of uniqueness of mean-flow-based linear analysis provides new opportunities for optimising models by specific choice of variables while also highlighting the importance of carefully accounting for the nonlinear terms that act as a forcing on the resolvent operator.

JFM classification

Acoustics: Jet noise

Information

Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , R5
Copyright
© The Author(s), 2020. Published by 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.)

Article purchase

Temporarily unavailable

References

REFERENCES

Akervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27 (5), 501–513.CrossRefGoogle Scholar
Balay, S., Gropp, W. D., McInnes, L. C. & Smith, B. F. 1997 Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing, pp. 163–202. Birkhäuser.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
Bres, G., Ham, F., Nichols, J. & Lele, S. 2017 Unstructured large-eddy simulations of supersonic jets. AIAA J. 55 (4), 1164–1184.CrossRefGoogle Scholar
Bres, 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.CrossRefGoogle Scholar
Bugeat, B., Chassaing, J.-C., Robinet, J.-C. & Sagaut, P. 2019 3d global optimal forcing and response of the supersonic boundary layer. J. Comput. Phys. 398, 108888.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., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.CrossRefGoogle Scholar
Chu, B.-T. 1965 On the energy transfer to small disturbances in fluid flow (part I). Acta Mechanica 1 (3), 215–234.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, 79–94.CrossRefGoogle Scholar
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397–413.CrossRefGoogle Scholar
Doak, P. E. 1995 Fluctuating total enthalpy as a generalized acoustic field. Acoust. Phys. 41, 677–685.Google Scholar
Goldstein, M. E. 2003 A generalized acoustic analogy. J. Fluid Mech. 488, 315–333.CrossRefGoogle Scholar
Hernández, V., Román, J. E., Tomás, A. & Vidal, V. 2007 Krylov–Schur methods in SLEPc. Tech Rep. STR-7. Universitat Politecnica de Valencia.Google Scholar
Hernandez, V., Roman, J. E. & Vidal, V. 2005 SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31 (3), 351–362.CrossRefGoogle Scholar
Howe, M. S. 1975 Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71 (4), 625–673.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Amplification of coherent streaks in the turbulent couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Jordan, P. & Colonius, T. 2013 Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45 (1), 173–195.CrossRefGoogle Scholar
Jordan, P. & Gervais, Y. 2008 Subsonic jet aeroacoustics: associating experiment, modelling and simulation. Exp. Fluids 44 (1), 1–21.CrossRefGoogle Scholar
Lesshafft, L., Huerre, P., Sagaut, P. & Terracol, M. 2006 Nonlinear global modes in hot jets. J. Fluid Mech. 554, 393–409.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, 063901.CrossRefGoogle Scholar
Lighthill, M. J. 1952 On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A 211 (1107), 564–587.Google Scholar
Lilley, G. M. 1974 On the noise from jets. AGARD CP-131, pp. 13–1.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.CrossRefGoogle ScholarPubMed
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Mettot, C., Renac, F. & Sipp, D. 2014 Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: application to open-loop control. J. Comput. Phys. 269, 234–258.CrossRefGoogle Scholar
Morra, P., Nogueira, P. A. S., Cavalieri, A. V. G. & Henningson, D. S. 2020 The colour of forcing statistics in resolvent analyses of turbulent channel flows. arXiv:2004.01565.Google Scholar
Morra, P., Semeraro, O., Henningson, D. S. & Cossu, C. 2019 On the relevance of reynolds stresses in resolvent analyses of turbulent wall-bounded flows. J. Fluid Mech. 867, 969–984.CrossRefGoogle Scholar
Pickering, E. M., Rigas, G., Sipp, D., Schmidt, O. T. & Colonius, T. 2019 Eddy viscosity for resolvent-based jet noise models. In 25th AIAA/CEAS Aeroacoustics Conference.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
Sinha, A., Rodríguez, D., Brès, G. A. & Colonius, T. 2014 Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 71–95.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar
Towne, A. 2016 Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets. PhD thesis, California Institute of Technology.Google Scholar
Towne, A., Brès, G. A. & Lele, S. K. 2016 Toward a resolvent-based statisitical jet-noise model. In Annual Research Briefs. Center for Turbulence Research, Stanford University.Google Scholar
Towne, A., Colonius, T., Jordan, P., Cavalieri, A. V. & Brès, G. A. 2015 Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet. In 21st AIAA/CEAS Aeroacoustics Conference.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, 821–867.CrossRefGoogle Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.CrossRefGoogle Scholar