Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T16:58:40.439Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic linear modes in a turbulent channel flow

Published online by Cambridge University Press:  17 February 2021

Gilles Tissot Open the ORCID record for Gilles Tissot [Opens in a new window] ,
André V. G. Cavalieri Open the ORCID record for André V. G. Cavalieri [Opens in a new window]  and
Étienne Mémin
Gilles Tissot*
Affiliation:
INRIA Rennes Bretagne Atlantique, IRMAR – UMR CNRS 6625, av. Général Leclerc, 35042Rennes, France
André V. G. Cavalieri
Affiliation:
Department of Aerospace Engineering, Instituto Tecnológico de Aeronáutica, Praça Mal. Eduardo Gomes 50, Vila das Acácias, 12228-900, São José dos Campos, Brazil
Étienne Mémin
Affiliation:
INRIA Rennes Bretagne Atlantique, IRMAR – UMR CNRS 6625, av. Général Leclerc, 35042Rennes, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This study is focused on the prediction of coherent structures, propagating within a turbulent channel flow. We propose a derivation of the linearised problem based on a stochastic formulation of the Navier–Stokes equations. It consists in considering the transport of quantities by a resolved velocity (i.e. solution of the model) perturbed by a Brownian motion which models the unresolved turbulent fluctuations over the time-averaged field, here thought of as the underlying background turbulence. The associated linearised model, considering the mean velocity profile as given, predicts linear solutions evolving within a corrected mean velocity field and perturbed by modelled background turbulence. Two ways to define the statistics of the Brownian motion are proposed and compared: one based on full simulation data, and the second, data free, based on preliminary predictions from resolvent analysis. The technique is applied on turbulent channel flows at friction Reynolds numbers $Re_\tau =180$ and $Re_\tau =550$, and predictions are compared with direct numerical simulation results. We show that the principal components of an ensemble of solutions of this stochastic linearised system are able to represent the leading spectral proper orthogonal decomposition modes with a similar accuracy to optimal responses coming from resolvent analysis with an eddy-viscosity model at scales where strong production occurs. For the other scales, receiving energy by nonlinear redistribution, the present strategy improves the prediction. Moreover, the second mode is systematically well predicted over all scales. This behaviour is understood by the ability of the stochastic modelling to model positive and negative inter-scale energy transfers through stochastic diffusion and random stochastic transport, while the eddy-viscosity term in resolvent analysis is purely diffusive.

JFM classification

Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 912 , 10 April 2021 , A51
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.)

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. 2020 b Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows. J. Fluid Mech. 900, A11.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. (EPL) 75 (5), 750–756.CrossRefGoogle Scholar
Bauer, W., Chandramouli, P., Chapron, B., Li, L. & Mémin, E. 2020 a Deciphering the role of small-scale inhomogeneity on geophysical flow structuration: a stochastic approach. J. Phys. Oceanogr. 50 (4), 983–1003.CrossRefGoogle Scholar
Bauer, W., Chandramouli, P., Li, L. & Mémin, E. 2020 b Stochastic representation of mesoscale eddy effects in coarse-resolution barotropic models. Ocean Model. 151, 1–50.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
Brynjell-Rahkola, M., Tuckerman, L.S., Schlatter, P. & Henningson, D.S. 2017 Computing optimal forcing using Laplace preconditioning. Commun. Comput. Phys. 22 (5), 1508–1532.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
Cavalieri, A.V.G., Rodriguez, D., Jordan, P., Colonius, T. & Gervais, Y. 2013 Wavepackets in the velocity field of turbulent jets. J. Fluid Mech. 730, 559–592.CrossRefGoogle Scholar
Cess, R.D. 1958 A survey of the literature on heat transfer in turbulent tubeflow. Research Report No. 8-0529-R24.Google Scholar
Chandramouli, P., Heitz, D., Laizet, S. & Mémin, E. 2018 Coarse large-eddy simulations in a transitional wake flow with flow models under location uncertainty. Comput. Fluids 168, 170–189.CrossRefGoogle Scholar
Chandramouli, P., Mémin, E. & Heitz, D. 2020 4D large scale variational data assimilation of a turbulent flow with a dynamics error model. J. Comput. Phys. 412, 109446.CrossRefGoogle Scholar
Chandramouli, P., Mémin, E., Heitz, D. & Fiabane, L. 2019 Fast 3D flow reconstructions from 2D cross-plane observations. Exp. Fluids 60 (2), 1–27.CrossRefGoogle Scholar
Chapron, B., Dérian, P., Mémin, E. & Resseguier, V. 2018 Large scale flows under location uncertainty: a consistent stochastic framework. Q. J. R. Meteorol. Soc. 144 (710), 251–260.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, 167–187.CrossRefGoogle Scholar
Del Álamo, J.C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), L41–L44.CrossRefGoogle Scholar
Del Álamo, J.C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Dergham, G., Sipp, D. & Robinet, J.-Ch. 2013 Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow. J. Fluid Mech. 719, 406–430.CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A: Fluid Dyn. 5 (11), 2600–2609.CrossRefGoogle Scholar
Farrell, B.F. & Ioannou, P.J. 1996 Generalized stability theory. Part I: autonomous operators. J. Atmos. Sci. 53 (14), 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Gibson, J.F., et al. 2019 Channelflow 2.0. Manuscript in preparation.Google Scholar
Gómez, F., Blackburn, H.M., Rudman, M., Sharma, A.S. & McKeon, B.J. 2016 a A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, R2.CrossRefGoogle Scholar
Gómez, F., Sharma, A.S. & Blackburn, H.M. 2016 b Estimation of unsteady aerodynamic forces using pointwise velocity data. J. Fluid Mech. 804, R4.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
Iyer, A.S., Witherden, F.D., Chernyshenko, S.I. & Vincent, P.E. 2019 Identifying eigenmodes of averaged small-amplitude perturbations to turbulent channel flow. J. Fluid Mech. 875, 758–780.CrossRefGoogle Scholar
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids (1994-present) 25 (10), 101302.CrossRefGoogle Scholar
Jovanović, M.R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kadri Harouna, S. & Mémin, E. 2017 Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling. Comput. Fluids 156, 456–469.CrossRefGoogle Scholar
Kaiser, T.L., Lesshafft, L. & Oberleithner, K. 2019 Prediction of the flow response of a turbulent flame to acoustic perturbations based on mean flow resolvent analysis. Trans. ASME: J. Engng Gas. Turbines Power 141 (11), 111021.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177 (1), 133–166.CrossRefGoogle Scholar
Kloeden, P.E. & Platen, E. 2013 Numerical Solution of Stochastic Differential Equations, Stochastic Modelling and Applied Probability, vol. 23. Springer Science & Business Media.Google Scholar
Leclercq, C., Demourant, F., Poussot-Vassal, C. & Sipp, D. 2019 Linear iterative method for closed-loop control of quasiperiodic flows. J. Fluid Mech. 868, 26–65.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
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
Luhar, M., Sharma, A.S. & McKeon, B.J. 2014 On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis. J. Fluid Mech. 751, 38–70.CrossRefGoogle Scholar
Martini, E., Cavalieri, A.V.G., Jordan, P., Towne, A. & Lesshafft, L. 2020 a Resolvent-based optimal estimation of transitional and turbulent flows. J. Fluid Mech. 900, A2.CrossRefGoogle Scholar
Martini, E., Rodríguez, D., Towne, A. & Cavalieri, A.V.G. 2020 b Efficient computation of global resolvent modes (preprint). arXiv:2008.10904.Google Scholar
McKeon, B.J. & Sharma, A.S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Mémin, E. 2014 Fluid flow dynamics under location uncertainty. Geophys. Astrophys. Fluid Dyn. 108 (2), 119–146.CrossRefGoogle Scholar
Moarref, R., Jovanović, M.R., Tropp, J.A., Sharma, A.S. & McKeon, B.J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids (1994-present) 26 (5), 051701.CrossRefGoogle Scholar
Moarref, R., Sharma, A.S., Tropp, J.A. & McKeon, B.J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.CrossRefGoogle 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.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., 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
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
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Pinier, B., Mémin, E., Laizet, S. & Lewandowski, R. 2019 Stochastic flow approach to model the mean velocity profile of wall-bounded flows. Phys. Rev. E 99 (6), 063101.CrossRefGoogle ScholarPubMed
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
Resseguier, V., Mémin, E. & Chapron, B. 2017 a Geophysical flows under location uncertainty, part I random transport and general models. Geophys. Astrophys. Fluid Dyn. 111 (3), 149–176.CrossRefGoogle Scholar
Resseguier, V., Mémin, E. & Chapron, B. 2017 b Geophysical flows under location uncertainty, part II quasi-geostrophy and efficient ensemble spreading. Geophys. Astrophys. Fluid Dyn. 111 (3), 177–208.CrossRefGoogle Scholar
Resseguier, V., Mémin, E. & Chapron, B. 2017 c Geophysical flows under location uncertainty, part III SQG and frontal dynamics under strong turbulence conditions. Geophys. Astrophys. Fluid Dyn. 111 (3), 209–227.CrossRefGoogle Scholar
Resseguier, V., Mémin, E., Heitz, D. & Chapron, B. 2017 d Stochastic modelling and diffusion modes for proper orthogonal decomposition models and small-scale flow analysis. J. Fluid Mech. 828, 29.Google Scholar
Reynolds, W.C. & Hussain, A.K.M.F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (2), 263–288.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
Ribeiro, J.H.M., Yeh, C. -A. & Taira, K. 2020 Randomized resolvent analysis. Phys. Rev. Fluids 5, 033902.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2001 Stability and Transition in Shear Flows. Springer-Verlag.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
Sharma, A.S., Moarref, R., McKeon, B.J., Park, J.S., Graham, M.D. & Willis, A.P. 2016 Low-dimensional representations of exact coherent states of the Navier–Stokes equations from the resolvent model of wall turbulence. Phys. Rev. E 93, 021102.CrossRefGoogle ScholarPubMed
Symon, S., Illingworth, S.J. & Marusic, I. 2020 Energy transfer in turbulent channel flows and implications for resolvent modelling (preprint). arXiv:2004.13266.CrossRefGoogle Scholar
Symon, S., Sipp, D. & McKeon, B.J. 2019 A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean. J. Fluid Mech. 881, 51–83.CrossRefGoogle Scholar
Towne, A., Lozano-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
Trefethen, L.N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Yang, Y. & Mémin, E. 2017 High-resolution data assimilation through stochastic subgrid tensor and parameter estimation from 4DEnVar. Tellus A 69 (1), 1308772.CrossRefGoogle Scholar
Zare, A., Georgiou, T.T. & Jovanović, M.R. 2019 Stochastic dynamical modeling of turbulent flows. Annu. Rev. Control Robot. Auton. Syst. 812, 636–680.Google Scholar
Zare, A., Jovanović, M.R. & Georgiou, T.T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.CrossRefGoogle Scholar