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Restricted nonlinear model for high- and low-drag events in plane channel flow

Published online by Cambridge University Press:  04 February 2019

Frédéric Alizard Open the ORCID record for Frédéric Alizard [Opens in a new window]  and
Damien Biau Open the ORCID record for Damien Biau [Opens in a new window]
Frédéric Alizard*
Affiliation:
LMFA, CNRS, Ecole Centrale de Lyon, Université Lyon 1, INSA Lyon, 43 Boulevard du 11 Novembre 1918, 69100 Villeurbanne, France
Damien Biau
Affiliation:
DynFluid Laboratory, Arts et Métiers ParisTech, 151 Boulevard de l’Hôpital, 75013 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

A restricted nonlinear (RNL) model, obtained by partitioning the state variables into streamwise-averaged quantities and superimposed perturbations, is used in order to track the exact coherent state in plane channel flow investigated by Toh & Itano (J. Fluid Mech., vol. 481, 2003, pp. 67–76). When restricting nonlinearities to quadratic interaction of the fluctuating part into the streamwise-averaged component, it is shown that the coherent structure and its dynamics closely match results from direct numerical simulation (DNS), even if only a single streamwise Fourier mode is retained. In particular, both solutions exhibit long quiescent phases, spanwise shifts and bursting events. It is also shown that the dynamical trajectory passes close to equilibria that exhibit either low- or high-drag states. When statistics are collected at times where the friction velocity peaks, the mean flow and root-mean-square profiles show the essential features of wall turbulence obtained by DNS for the same friction Reynolds number. For low-drag events, the mean flow profiles are related to a universal asymptotic state called maximum drag reduction (Xi & Graham, Phys. Rev. Lett., vol. 108, 2012, 028301). Hence, the intermittent nature of self-sustaining processes in the buffer layer is contained in the dynamics of the RNL model, organized in two exact coherent states plus an asymptotic turbulent-like attractor. We also address how closely turbulent dynamics approaches these equilibria by exploiting a DNS database associated with a larger domain.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 221 - 243
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Copyright
© 2019 Cambridge University Press 

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References

Abe, H., Antonia, R. A. & Toh, S. 2018 Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit. J. Fluid Mech. 850, 733–768.Google Scholar
Alfredsson, P. H., Örlü, R. & Schlatter, P. 2011 The viscous sublayer revisited – exploiting self-similarity to determine the wall position and friction velocity. Exp. Fluids 51 (1), 271–280.Google Scholar
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27, 105103.Google Scholar
Alizard, F. 2017 Invariant solutions in a channel flow using a minimal restricted nonlinear model. C. R. Méc. 345, 117–124.Google Scholar
Biau, D. & Bottaro, A. 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc. Lond. A 367, 529–544.Google Scholar
Blackburn, H. M., Hall, P. & Sherwin, S. J. 2013 Lower branch equilibria in couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726, R2.Google Scholar
Bretheim, J. U., Meneveau, C. & Gayme, D. F. 2015 Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel. Phys. Fluids 27, 011702.Google Scholar
Bretheim, J. U., Meneveau, C. & Gayme, D. F. 2018 A restricted nonlinear large eddy simulation model for high Reynolds number flows. J. Turbul. 19, 141–166.Google Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.Google Scholar
Cossu, C. & Hwang, Y. 2017 Self-sustaining processes at all scales in wall-bounded turbulent shear flows. Phil. Trans. R. Soc. Lond. A 375, 1–14.Google Scholar
Curry, J. H., Herring, J. R., Loncaric, J. & Orszag, S. A. 1984 Order and disorder in two- and three-dimensional Bénard convection. J. Fluid Mech. 147, 1–38.Google Scholar
Diwan, S. S. & Morrison, J. F. 2017 Spectral structure and linear mechanisms in a rapidly distorted boundary layer. Intl J. Heat Fluid Flow 67, Part B, 63–73.Google Scholar
Duguet, Y., Willis, A. P. & Kerswell, R. R. 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255–274.Google Scholar
Eckhardt, B. 2014 Doubly localized states in plane Couette flow. J. Fluid Mech. 758, 1–4.Google Scholar
Farrell, B. F., Gayme, D. F. & Ioannou, P. J. 2017 A statistical state dynamics approach to wall turbulence. Phil. Trans. R. Soc. Lond. A 375 (2089), 20160081.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II: nonautonomous operators. J. Atmos. Sci. 53 (14), 2041–2053.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.Google Scholar
Farrell, B. F., Ioannou, P. J., Jiménez, J., Constantinou, N. C., Lozano-Durán, A. & Nikolaidis, M. A. 2016 A statistical state dynamics based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 290–315.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.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.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the problems of turbulence. J. Fluid Mech. 212, 497–532.Google Scholar
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in turbulent channel flow. J. Fluid Mech. 664, 51–73.Google Scholar
Hwang, Y., Willis, A. P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for Re 𝜏 up to 1000. J. Fluid Mech. 802, R1.Google Scholar
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703–716.Google Scholar
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.Google Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.Google Scholar
Kawahara, G., Uhlmann, M. & Veen, L. V. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.Google Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, 1–11.Google Scholar
Kim, H. T., Kline, S. J. & Reynolds, W. C. 1971 The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50 (1), 133–160.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (4), 741–773.Google Scholar
Kreilos, T., Veble, G., Schneider, T. M. & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100–122.Google Scholar
Kushwaha, A., Park, J. S. & Graham, M. D. 2017 Temporal and spatial intermittencies within channel flow turbulence near transition. Phys. Rev. Fluids 2, 024603.Google Scholar
Landhal, M. T. 1980 A note on an algebraic instability of invscid parallel shear flow. J. Fluid Mech. 98, 243–251.Google Scholar
Lee, M. J. & Moin, J. K. P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.Google Scholar
Limpert, E., Stahel, W. A. & Abbt, M. 2001 Log-normal distributions across the sciences: keys and clues. BioScience 51 (5), 341–352.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.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
Örlü, R. & Schlatter, P. 2011 On the fluctuating wall-shear stress in zero pressure-gradient turbulent boundary layer flows. Phys. Fluids 23 (2), 021704.Google Scholar
Panton, R. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37, 341–383.Google 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.Google Scholar
Pausch, M., Yang, Q., Hwang, Y. & Eckhardt, B. 2019 Quasilinear approximation for exact coherent states in parallel shear flows. Fluid Dyn. Res. 51, 011402.Google Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.Google Scholar
Ragone, F., Wouters, J. & Bouchet, F. 2017 Computation of extreme heat waves in climate models using a large deviation algorithm. Proc. Natl Acad. Sci. 115 (1), 24–29.Google Scholar
Rao, K. N., Narasimha, R. & Narayanan, M. A. B. 1971 The bursting phenomenon in a turbulent boundary layer. J. Fluid Mech. 48 (2), 339–352.Google Scholar
Rawat, S., Cossu, C. & Rincon, F. 2014 Relative periodic orbits in plane Poiseuille flow. C. R. Méc. 342, 485–489.Google Scholar
Rawat, S., Cossu, C. & Rincon, F. 2016 Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow. C. R. Méc. 344, 448–455.Google Scholar
Rinaldi, E., Schlatter, P. & Bagheri, S. 2018 Edge state modulation by mean viscosity gradients. J. Fluid Mech. 838, 379–403.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601–639.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.Google Scholar
Schneider, T. M., Gisbon, J. F., Lagha, M., Lillo, F. D. & Eckhardt, B. 2008 Laminar–turbulent boundary in plane Couette flow. Phys. Rev. E 78, 037301.Google Scholar
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010 Localized edge states nucleate turbulence in extended plane couette cells. J. Fluid Mech. 646, 441–451.Google Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.Google Scholar
Sharma, A. S., Moarref, R. & McKeon, B. J. 2017 Scaling and interaction of self-similar modes in models of high Reynolds number wall turbulence. Phil. Trans. R. Soc. Lond. A 375, 1–14.Google 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(R).Google Scholar
Smith, C. R. & Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27–54.Google Scholar
Tailleur, J. & Kurchan, J. 2007 Probing rare physical trajectories with Lyapunov weighted dynamics. Nat. Phys. 3, 203–207.Google Scholar
Taylor, G. I. 1935 Turbulence in a contracting stream. Z. Angew. Math. Mech. 15, 91–96.Google Scholar
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 67–76.Google Scholar
Toh, S. & Itano, T. 2005 Interaction between a large-scale structure and near-wall structures in channel flow. J. Fluid Mech. 524, 249–262.Google Scholar
Virk, P., Mickley, H. & Smith, K. 1970 The ultimate asymptote and mean flow structure in Toms phenomenon. Trans. ASME E: J. Appl. Mech. 37, 488–493.Google Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Waleffe, F. & Kim, J. 1997 How Streamwise Rolls and Streaks Self-Sustain in a Shear Flow (ed. Panton, R.), pp. 309–332. Computational Mechanics Publications.Google Scholar
Wallace, J. M., Eckelmann, H. & Brodkey, R. S. 1972 The wall region in turbulent shear flow. J. Fluid Mech. 54 (1), 39–48.Google Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.Google Scholar
Willis, A. P. & Kerswell, R. R. 2009 Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states. J. Fluid Mech. 619, 213–233.Google Scholar
Xi, L. & Bai, X. 2016 Marginal turbulent state of viscoelastic fluids: a polymer drag reduction perspective. Phys. Rev. E 93, 043118.Google Scholar
Xi, L. & Graham, M. D. 2012 Dynamics on the laminar–turbulent boundary and the origin of the maximum drag reduction asymptote. Phys. Rev. Lett. 108, 028301.Google Scholar
Zammert, S. & Eckhardt, B. 2014 Periodically bursting edge states in plane Poiseuille flow. Fluid. Dyn. Res. 46, 041419.Google Scholar