Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T21:08:26.916Z Has data issue: false hasContentIssue false
  • English
  • Français

A sparse optimal closure for a reduced-order model of wall-bounded turbulence

Published online by Cambridge University Press:  24 March 2022

Zhao Chua Khoo ,
Chi Hin Chan Open the ORCID record for Chi Hin Chan [Opens in a new window]  and
Yongyun Hwang Open the ORCID record for Yongyun Hwang [Opens in a new window]
Zhao Chua Khoo
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
Chi Hin Chan*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
Yongyun Hwang
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present study, a set of physics-informed and data-driven approaches are examined towards the development of an accurate reduced-order model for a turbulent plane Couette flow. Based on the utilisation of the proper orthogonal decomposition (POD), a particular focus is given to the development of a reduced-order model where the number of POD modes are not large enough to cover the full dynamics of the given turbulent state, the situation directly relevant to the reduced-order modelling for turbulent flows. Starting from the conventional Galerkin projection approach ignoring the truncation error, three approaches enhanced by both physics and data are examined: (1) sparse regression of the POD-Galerkin dynamics; (2) Galerkin projection with an empirical eddy-viscosity model; (3) Galerkin projection with an optimal eddy viscosity obtained from a newly proposed sparse regression – an idea applying the sparse identification of nonlinear dynamics framework to an eddy-viscosity model. The sparse regression of the POD-Galerkin dynamics does not perform well, as the number of POD modes for the given chaotic dynamics appears to be too small. While the unsatisfactory performance of the Galerkin projection model with an empirical eddy viscosity is observed, the newly proposed approach, which combines the concept of an optimal eddy-viscosity closure with a sparse regression, more accurately approximates the chaotic dynamics than the other reduced-order models considered. This is demonstrated with the mean and time scales of the POD mode amplitudes as well as the first- and second-order turbulence statistics.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 939 , 25 May 2022 , A11
Check for updates
Copyright
© The Author(s), 2022. 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

del Álamo, J.C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15, L41.CrossRefGoogle Scholar
del Álamo, J.C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27, 105103.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J.L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192–196.CrossRefGoogle ScholarPubMed
Barkley, D. 2016 Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech. 803, P1.CrossRefGoogle Scholar
Beetham, S. & Capecelatro, J. 2020 Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids 5 (8), 084611.CrossRefGoogle Scholar
Beetham, S., Fox, R.O. & Capecelatro, J. 2021 Sparse identification of multiphase turbulence closures for coupled fluid–particle flows. J. Fluid Mech. 914, A11.CrossRefGoogle Scholar
Bewley, T.R., Moin, P. & Temam, R. 2001 Dns-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179–225.CrossRefGoogle Scholar
Brunton, S.L. & Kutz, J.N. 2019 Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press.CrossRefGoogle Scholar
Brunton, S.L., Noack, B.R. & Koumoutsakos, P. 2020 Machine learning for fluid mechanics. Annu. Rev. Fluid Mech. 52, 477–508.CrossRefGoogle Scholar
Brunton, S.L., Proctor, J.L. & Kutz, J.N. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl Acad. Sci. USA 113 (15), 3932–3937, https://www.pnas.org/content/113/15/3932.full.pdf.CrossRefGoogle ScholarPubMed
Butler, K.M. & Farrell, B.F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.CrossRefGoogle Scholar
Callaham, J.L., Brunton, S.L. & Loiseau, J.-C. 2021 a On the role of nonlinear correlations in reduced-order modeling. arXiv:2106.02409.CrossRefGoogle Scholar
Callaham, J.L., Loiseau, J.-C., Rigas, G. & Brunton, S.L. 2021 b Nonlinear stochastic modelling with langevin regression. Proc. R. Soc. Lond. A 477 (2250), 487–505.Google ScholarPubMed
Cassinelli, A., de Giovanetti, M. & Hwang, Y. 2017 Streak instability in near-wall turbulence revisited. J. Turbul. 18 (5), 443–464.CrossRefGoogle Scholar
Cavalieri, A.V.G. 2021 Structure interactions in a reduced-order model for wall-bounded turbulence. Phys. Rev. Fluids 6, 034610.CrossRefGoogle Scholar
Chantry, M., Tuckerman, L.S. & Barkley, D. 2017 Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech. 824, R1.CrossRefGoogle Scholar
Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474–504.CrossRefGoogle Scholar
Choi, H., Temam, R., Moin, P. & Kim, J. 1993 Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech. 245, 509–543.CrossRefGoogle Scholar
Cordier, L., El Majd, B.A. & Favier, J. 2010 Calibration of POD reduced-order models using Tikhonov regularization. Intl J. Numer. Meth. Fluids 63 (2), 269–296.Google 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
Couplet, M., Basdevant, C. & Sagaut, P. 2005 Calibrated reduced-order POD-Galerkin system for fluid flow modelling. J. Comput. Phys. 207 (1), 192–220.CrossRefGoogle Scholar
Couplet, M., Sagaut, P. & Basdevant, C. 2003 Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow. J. Fluid Mech. 491 (491), 275–284.CrossRefGoogle Scholar
Crisanti, A., Jensen, M.H., Paladin, G. & Vulpiani, A. 1993 Predictability of velocity and temperature fields in intermittent turbulence. J. Phys. A 26, 6943.CrossRefGoogle Scholar
Doohan, P., Bengana, Y., Yang, Q., Willis, A.P. & Hwang, Y. 2022 Multi-scale state space and travelling waves in wall-bounded turbulence. J. Fluid Mech. (submitted).Google Scholar
Doohan, P., Willis, A.P. & Hwang, Y. 2019 Shear stress-driven flow: the state space of near-wall turbulence as $Re_\tau \rightarrow \infty$. J. Fluid Mech. 874, 606–638.CrossRefGoogle Scholar
Doohan, P., Willis, A.P. & Hwang, Y. 2021 Minimal multi-scale dynamics of near-wall turbulence. J. Fluid Mech. 913, A8.CrossRefGoogle Scholar
Duraisamy, K. 2021 Perspectives on machine learning-augmented reynolds-averaged and large eddy simulation models of turbulence. Phys. Rev. Fluids 6, 050504.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T.M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447.CrossRefGoogle Scholar
Faisst, H. & Eckhardt, B. 2003 Travelling waves in pipe flow. Phys. Rev. Lett. 91, 224502.CrossRefGoogle Scholar
Falco, R.E. 1977 Coherent motions in the outer region of turbulent boundary layers. Phys. Fluids 20, S124–S132.CrossRefGoogle Scholar
Farazmand, M. 2016 An adjoint-based approach for finding invariant solutions of Navier–Stokes equations. J. Fluid Mech. 795, 278–312.CrossRefGoogle Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.CrossRefGoogle Scholar
Gelß, P., Klus, S., Eisert, J. & Schütte, C. 2019 Multidimensional approximation of nonlinear dynamical systems. J. Comput. Nonlinear Dyn. 14 (6), 061006.Google Scholar
Gibson, J.F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gibson, J.F., Halcrow, J. & Cvitanovic, P. 2009 Equilibrium and traveling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
de Giovanetti, M., Sung, H.J. & Hwang, Y. 2017 Streak instability in turbulent channel flow: the seeding mechanism of large-scale motions. J. Fluid Mech. 832, 483–513.CrossRefGoogle Scholar
Graham, M.D. & Floryan, D. 2020 Exact coherent states and the nonlinear dynamics of wall-bounded turbulent flows. Annu. Rev. Fluid Mech. 53, 227–253.CrossRefGoogle Scholar
Hall, P. & Sherwin, S.J. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hamilton, J.M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287 (1), 317–348.CrossRefGoogle Scholar
Head, M.R. & Bandyopadhay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.CrossRefGoogle Scholar
Holmes, P., Lumley, J.L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 723, 264–288.CrossRefGoogle Scholar
Hwang, Y. & Bengana, Y. 2016 Self-sustaining process of minimal attached eddies in turbulent channel flow. J. Fluid Mech. 795, 708–738.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 a 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
Hwang, Y. & Cossu, C. 2010 b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 c Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105, 044505.CrossRefGoogle ScholarPubMed
Hwang, Y. & Cossu, C. 2011 Self-sustained processes in the logarithmic layer of turbulent channel flows. Phys. Fluids 23, 061702.CrossRefGoogle Scholar
Hwang, Y. & Lee, M. 2020 The mean logarithm emerges with self-similar energy balance. J. Fluid Mech. 903, R6.CrossRefGoogle Scholar
Hwang, Y., Willis, A.P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for $Re_\tau$ up to 1000. J. Fluid Mech. 802, R1.CrossRefGoogle Scholar
Jeong, J., Benney, F., Schoppa, W. & Kim, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185–214.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.CrossRefGoogle Scholar
Jiménez, J. & Simens, M.P. 2001 Low-dimensional dynamics of a turbulent wall flow. J. Fluid Mech. 435, 81–91.CrossRefGoogle Scholar
Joseph, D.D. 1976 Stability of Fluid Motions. Springer Tracts in Natural Philosophy, vol. 27. Springer.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Kim, K.C. & Adrian, R. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.CrossRefGoogle Scholar
Kline, S.J., Reynolds, W.C., Schraub, F.A. & Runstadler, P.W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.CrossRefGoogle Scholar
Kovasznay, L.S.G., Kibens, V. & Blackwelder, R.F. 1970 Large-scale motion in the intermittent region of a turbulent boundary layer. J. Fluid Mech. 41, 283–325.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.CrossRefGoogle ScholarPubMed
Lagha, M. & Manneville, P. 2007 Modelling transitional plane Couette flow. Eur. Phys. J. B 58, 433–447.CrossRefGoogle Scholar
Lee, C., Kim, J. & Choi, H. 1999 Suboptimal control of turbulent channel flow for drag reduction. J. Fluid Mech. 358, 245–258.CrossRefGoogle Scholar
Lee, M.K. & Moser, R.D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886–938.CrossRefGoogle Scholar
Loiseau, J.-C. & Brunton, S.L. 2018 Constrained sparse galerkin regression. J. Fluid Mech. 838, 42–67.CrossRefGoogle Scholar
Lozano-Durán, A., Nikolaidis, M.-A., Constantinou, N.C. & Karp, M. 2021 Cause-and-effect of linear mechanisms sustaining wall turbulence. J. Fluid Mech. 914, A8.CrossRefGoogle Scholar
Lumley, J.L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation (ed. A.M. Yaglom & V.I. Tatarski), Nonlinear Problems of Fluid Dynamics. Nakua.Google Scholar
Lumley, J.L. 1981 Coherent structures in turbulence. In Transition and Turbulence (ed. Richard E. Meyer), pp. 215–242. Elsevier.CrossRefGoogle Scholar
Marusic, I. & Monty, J.P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2004 A low-dimensional model for turbulent shear flows. New J. Phys. 6, 56.CrossRefGoogle Scholar
Moehlis, J., Faisst, H. & Eckhardt, B. 2005 Periodic orbits and chaotic sets in a low-dimensional model for shear flows. SIAM J. Appl. Dyn. Syst. 4, 352–376.CrossRefGoogle Scholar
Mohebujjaman, M., Rebholz, L.G. & Iliescu, T. 2019 Physically constrained data-driven correction for reduced-order modeling of fluid flows. Intl J. Numer. Meth. Fluids 89 (3), 103–122.CrossRefGoogle Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Noack, B.R., Morzýnski, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control. CISM Courses and Lectures. Springer.CrossRefGoogle Scholar
Östh, J., Noack, B.R., Krajnović, S., Barros, D. & Borée, J. 2014 On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an ahmed body. J. Fluid Mech. 747, 518–544.CrossRefGoogle Scholar
Park, J., Hwang, Y. & Cossu, C. 2011 On the stability of large-scale streaks in the turbulent Couette and Poiseuille flows. C. R. Mèc 339 (1), 1–5.CrossRefGoogle Scholar
Park, J.S. & Graham, M.D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430–454.CrossRefGoogle Scholar
Pathak, J., Hunt, B., Girvan, M., Lu, Z. & Ott, E. 2018 Model-free prediction of large spatiotemporally chaotic systems from data: a reservoir computing approach. Phys. Rev. Lett. 120, 024102.CrossRefGoogle ScholarPubMed
Podvin, B. 2009 A proper-orthogonal-decomposition–based model for the wall layer of a turbulent channel flow. Phys. Fluids 21 (1), 015111.CrossRefGoogle Scholar
Protas, B., Noack, B.R. & Östh, J. 2015 Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows. J. Fluid Mech. 766, 337–367.CrossRefGoogle Scholar
Rawat, S., Cossu, C., Hwang, Y. & Rincon, F. 2015 On the self-sustained nature of large-scale motions in turbulent Couette flow. J. Fluid Mech. 782, 515–540.CrossRefGoogle Scholar
Rempfer, D. & Fasel, F.H. 1994 a Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257–283.CrossRefGoogle Scholar
Rempfer, D. & Fasel, F.H. 1994 b Evolution of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 260, 351–375.CrossRefGoogle Scholar
Rubini, R., Lasagna, D. & Da Ronch, A. 2021 The $l_1$-based sparsification of energy interactions in unsteady lid-driven cavity flow. J. Fluid Mech. 905, A18.Google Scholar
Ruelle, D. 1979 Microscopic fluctuations and turbulence. Phys. Lett. A 72, 81.CrossRefGoogle Scholar
Schmelzer, M., Dwight, R.P. & Cinnella, P. 2020 Discovery of algebraic Reynolds-stress models using sparse symbolic regression. Flow Turbul. Combust. 104 (2), 579–603.CrossRefGoogle Scholar
Schmidt, M. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 264, 255–275.Google Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures; part I: coherent structures. Q. Appl. Maths XLV (3), 561–571.CrossRefGoogle Scholar
Smith, T.R., Moehlis, J. & Holmes, Ph. 2005 Low-dimensional models for turbulent plane Couette flow in a minimal flow unit. J. Fluid Mech. 538, 71–110.CrossRefGoogle Scholar
Townsend, A.A. 1956 The Structure of Turbulent Shear Flow, 1st edn. Cambridge University Press.Google Scholar
Townsend, A.A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Vassilicos, J.C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47, 95–114.CrossRefGoogle Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Waleffe, F. 1995 Hydrodynamic stability and turbulence: beyond transients to a self-sustaining process. Stud. Appl. Maths 95, 319–343.CrossRefGoogle Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.CrossRefGoogle Scholar
Wedin, H. & Kerswell, R.R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Willis, A.P., Cvitanovic, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514–540.CrossRefGoogle Scholar
Yang, Q., Willis, A.P. & Hwang, Y. 2019 Exact coherent states of attached eddies in channel flow. J. Fluid Mech. 862, 1029–1059.CrossRefGoogle Scholar