Hostname: page-component-669899f699-tzmfd Total loading time: 0 Render date: 2025-05-06T06:35:44.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Resolvent-based estimation of turbulent channel flow using wall measurements

Published online by Cambridge University Press:  24 September 2021

Filipe R. Amaral Open the ORCID record for Filipe R. Amaral [Opens in a new window] ,
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Eduardo Martini Open the ORCID record for Eduardo Martini [Opens in a new window] ,
Peter Jordan Open the ORCID record for Peter Jordan [Opens in a new window]  and
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]
Filipe R. Amaral*
Affiliation:
Divisão de Engenharia Aeronáutica, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
André V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeronáutica, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
Eduardo Martini
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS–Université de Poitiers–ENSMA, 86000 Poitiers, France
Peter Jordan
Affiliation:
Département Fluides, Thermique, Combustion, Institut Pprime, CNRS–Université de Poitiers–ENSMA, 86000 Poitiers, France
Aaron Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We employ a resolvent-based methodology to estimate velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of approximately 180, 550 and 1000 using measurements of shear stress and pressure at the walls, taken from direct numerical simulation (DNS) databases. Martini et al. (J. Fluid Mech., vol. 900, 2021, p. A2) showed that the resolvent-based estimator is optimal when the true space–time forcing statistics are utilised, thus providing an upper bound for the accuracy of any linear estimator. We use this framework to determine the flow structures that can be linearly estimated from wall measurements, and we characterise these structures and the estimation errors in both physical and wavenumber space. We also compare these results to those obtained using approximate forcing models – an eddy-viscosity model and white-noise forcing – and demonstrate the significant benefit of using true forcing statistics. All models lead to accurate results up to the buffer layer, but only using the true forcing statistics allows accurate estimation of large-scale logarithmic-layer structures, with significant correlation between the estimates and DNS results throughout the channel. The eddy-viscosity model displays an intermediate behaviour, which may be related to its ability to partially capture the forcing colour. Our results show that structures that leave a footprint on the channel walls can be accurately estimated using the linear resolvent-based methodology, and the presence of large-scale wall-attached structures enables accurate estimations through the logarithmic layer.

JFM classification

Waves/Free-surface Flows: Channel flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 927 , 25 November 2021 , A17
Check for updates
Copyright
© The Author(s), 2021. 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

Abreu, L.I., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020 a Resolvent modelling of near-wall coherent structures in turbulent channel flow. Intl J. Heat Fluid Flow 85, 108662.CrossRefGoogle Scholar
Abreu, L.I., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020b Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows. J. Fluid Mech. 900, A11.CrossRefGoogle Scholar
Abreu, L.I., Cavalieri, A.V.G. & Wolf, W. 2017 Coherent hydrodynamic waves and trailing-edge noise. AIAA Paper 2017-3173.CrossRefGoogle Scholar
Anantharamu, S. & Mahesh, K. 2020 Analysis of wall-pressure fluctuation sources from direct numerical simulation of turbulent channel flow. J. Fluid Mech. 898, A17.CrossRefGoogle Scholar
Bagheri, S., Henningson, D.S., Hoepffner, J. & Schmid, P.J. 2009 Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62 (2), 020803.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
Bewley, T.R. & Protas, B. 2004 Skin friction and pressure: the ‘footprints’ of turbulence. Physica D 196 (1–2), 28–44.CrossRefGoogle Scholar
Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71, 020802.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Bewley, T.R. & Henningson, D.S. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552 (1), 167.CrossRefGoogle Scholar
Colburn, C.H., Cessna, J.B. & Bewley, T.R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289–303.CrossRefGoogle Scholar
Del Álamo, J.C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41.CrossRefGoogle Scholar
Del Álamo, J.C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205.CrossRefGoogle Scholar
Del Álamo, J.C., Jiménez, J., Zandonade, P. & Moser, R.D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135.CrossRefGoogle Scholar
Encinar, M.P. & Jiménez, J. 2019 Logarithmic-layer turbulence: a view from the wall. Phys. Rev. Fluids 4 (11), 114603.CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149.CrossRefGoogle Scholar
Gibson, J.F., et al. 2019 Channelflow 2.0. Available at: https://www.channelflow.ch/.Google Scholar
Guastoni, L., Güemes, A., Ianiro, A., Discetti, S., Schlatter, P., Azizpour, H. & Vinuesa, R. 2020 Convolutional-network models to predict wall-bounded turbulence from wall quantities. arXiv:2006.12483.Google Scholar
Hœpffner, J., Chevalier, M., Bewley, T.R. & Hennington, D.S. 2005 State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows. J. Fluid Mech. 534, 263–294.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
Illingworth, S.J., Monty, J.P. & Marusic, I. 2018 Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146–162.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. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.CrossRefGoogle Scholar
Jung, J., Martini, E., Cavalieri, A., Jordan, P., Lesshafft, L. & Towne, A. 2020 Optimal resolvent-based estimation for flow control. Bull. Am. Phys. Soc. 65 (13), G06.00004.Google Scholar
Lee, M. & Moser, R.D. 2015 Direct numerical simulation of turbulent channel flow up to $Re_{\tau } \approx 5200$. J. Fluid Mech. 774, 395–415.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
Lozano-Durán, A. & Jiménez, J. 2014 Effect of the computational domain on direct simulations of turbulent channels up to $Re_{\tau }=4200$. Phys. Fluids 26 (1), 011702.CrossRefGoogle Scholar
Martinelli, F. 2009 Feedback control of turbulent wall flows. PhD thesis, Politecnico di Milano.Google Scholar
Martini, E. 2019 Linear modelling of flows: physical mechanisms and tools. PhD thesis, Instituto Tecnológico de Aeronáutica.Google Scholar
Martini, E., Jordan, P., Cavalieri, A.V.G., Towne, A. & Lesshafft, L. 2020 Resolvent-based optimal estimation of transitional and turbulent flows. J. Fluid Mech. 900, A2.CrossRefGoogle Scholar
Marusic, I., Baars, W.J. & Hutchins, N. 2017 Scaling of the streamwise turbulence intensity in the context of inner-outer interactions in wall turbulence. Phys. Rev. Fluids 2 (10), 100502.CrossRefGoogle Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 Predictive model for wall-bounded turbulent flow. Science 329 (5988), 193–196.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
Meditch, J.S. 1973 A survey of data smoothing for linear and nonlinear dynamic systems. Automatica 9 (2), 151–162.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
Morra, P., Sasaki, K., Hanifi, A., Cavalieri, A.V.G. & Henningson, D.S. 2020 A realizable data-driven approach to delay bypass transition with control theory. J. Fluid Mech. 883, A33.CrossRefGoogle 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
Oehler, S., Garcia-Gutiérrez, A. & Illingworth, S. 2018 Linear estimation of coherent structures in wall-bounded turbulence at $Re_{\tau }=2000$. J. Phys.: Conf. Ser. 1001 (1), 012006.Google Scholar
Pickering, E., Rigas, G., Schmidt, O.T., Sipp, D. & Colonius, T. 2021 Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets. J. Fluid Mech. 917, A29.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.CrossRefGoogle Scholar
Reynolds, W.C. & Tiederman, W.G. 1967 Stability of turbulent channel flow, with application to Malkus's theory. J. Fluid Mech. 27 (2), 253–272.CrossRefGoogle Scholar
Sasaki, K., Piantanida, S., Cavalieri, A.V.G. & Jordan, P. 2017 Real-time modelling of wavepackets in turbulent jets. J. Fluid Mech. 821, 458–481.CrossRefGoogle Scholar
Sasaki, K., Vinuesa, R., Cavalieri, A.V.G., Schlatter, P. & Henningson, D.S. 2019 Transfer functions for flow predictions in wall-bounded turbulence. J. Fluid Mech. 864, 708–745.CrossRefGoogle Scholar
Schmidt, O.T., Towne, A., Rigas, G., Colonius, T. & Brés, G.A. 2018 Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982.CrossRefGoogle Scholar
Sharma, A.S. & McKeon, B.J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Smits, A.J., McKeon, B.J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.CrossRefGoogle Scholar
Symon, S., Illingworth, S.J. & Marusic, I. 2021 Energy transfer in turbulent channel flows and implications for resolvent modelling. J. Fluid Mech. 911, A3.CrossRefGoogle Scholar
Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V. & Ukeiley, L.S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 4013–4041.CrossRefGoogle Scholar
Thomas, V.L., Farrell, B.F., Ioannou, P.J. & Gayme, D.F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27 (10), 105104.CrossRefGoogle Scholar
Tinney, C.E., Coiffet, F., Delville, J., Hall, A.M., Jordan, P. & Glauser, M.N. 2006 On spectral linear stochastic estimation. Exp. Fluids 41 (5), 763–775.CrossRefGoogle 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.CrossRefGoogle Scholar
Tol, H.J., de Visser, C.C. & Kotsonis, M. 2019 Experimental model-based estimation and control of natural Tollmien–Schlichting waves. AIAA J. 57 (6), 2344–2355.CrossRefGoogle Scholar
Towne, A., Brés, G.A. & Lele, S.K. 2017 A statistical jet-noise model based on the resolvent framework. AIAA Paper 2017-3706.CrossRefGoogle Scholar
Towne, A., Lozando-Durán, A. & Yang, X. 2020 Resolvent-based estimation of space–time flow statistics. J. Fluid Mech. 883, A17.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
Welch, P.D. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 70–73.CrossRefGoogle Scholar

Amaral et al. supplementary movie 1

See word file for movie caption

Download Amaral et al. supplementary movie 1(Video)
Video 7.2 MB

Amaral et al. supplementary movie 2

See word file for movie caption

Download Amaral et al. supplementary movie 2(Video)
Video 4.4 MB

Amaral et al. supplementary movie 3

See word file for movie caption

Download Amaral et al. supplementary movie 3(Video)
Video 4.8 MB

Amaral et al. supplementary movie 4

See word file for movie caption

Download Amaral et al. supplementary movie 4(Video)
Video 1.5 MB

Amaral et al. supplementary movie 5

See word file for movie caption

Download Amaral et al. supplementary movie 5(Video)
Video 1.9 MB
Supplementary material: File

Amaral et al. supplementary material

Captions for movies 1-5

Download Amaral et al. supplementary material(File)
File 2.8 KB