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

  • Frédéric Alizard (a1) and Damien Biau (a2)


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.


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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, 733768.10.1017/jfm.2018.434
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), 271280.10.1007/s00348-011-1048-8
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27, 105103.10.1063/1.4932178
Alizard, F. 2017 Invariant solutions in a channel flow using a minimal restricted nonlinear model. C. R. Méc. 345, 117124.10.1016/j.crme.2016.11.005
Biau, D. & Bottaro, A. 2009 An optimal path to transition in a duct. Phil. Trans. R. Soc. Lond. A 367, 529544.10.1098/rsta.2008.0191
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, R2.10.1017/jfm.2013.254
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, 011702.10.1063/1.4906987
Bretheim, J. U., Meneveau, C. & Gayme, D. F. 2018 A restricted nonlinear large eddy simulation model for high Reynolds number flows. J. Turbul. 19, 141166.10.1080/14685248.2017.1403031
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774777.10.1063/1.858663
Cossu, C. & Hwang, Y. 2017 Self-sustaining processes at all scales in wall-bounded turbulent shear flows. Phil. Trans. R. Soc. Lond. A 375, 114.10.1098/rsta.2016.0088
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, 138.10.1017/S0022112084001968
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, 63–73.10.1016/j.ijheatfluidflow.2017.04.009
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, 255274.10.1017/S0022112008003248
Eckhardt, B. 2014 Doubly localized states in plane Couette flow. J. Fluid Mech. 758, 14.10.1017/jfm.2014.442
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), 20160081.10.1098/rsta.2016.0081
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II: nonautonomous operators. J. Atmos. Sci. 53 (14), 20412053.10.1175/1520-0469(1996)053<2041:GSTPIN>2.0.CO;2
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149196.10.1017/jfm.2012.300
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, 290315.10.1017/jfm.2016.661
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 2561.10.1017/jfm.2014.89
Gibson, J. F., Halcrow, J. & Cvitanovic, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.10.1017/S002211200800267X
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.10.1017/S0022112010002892
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.10.1017/S0022112095000978
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the problems of turbulence. J. Fluid Mech. 212, 497532.10.1017/S0022112090002075
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in turbulent channel flow. J. Fluid Mech. 664, 5173.10.1017/S0022112010003629
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, R1.10.1017/jfm.2016.470
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703716.10.1143/JPSJ.70.703
Jiménez, J. 2018 Coherent structures in wall-bounded turbulence. J. Fluid Mech. 842, P1.10.1017/jfm.2018.144
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.10.1017/S0022112091002033
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335359.10.1017/S0022112099005066
Kawahara, G., Uhlmann, M. & Veen, L. V. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.10.1146/annurev-fluid-120710-101228
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, 111.10.1017/jfm.2013.20
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), 133160.10.1017/S0022112071002490
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (4), 741773.10.1017/S0022112067001740
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, 100122.10.1017/jfm.2013.212
Kushwaha, A., Park, J. S. & Graham, M. D. 2017 Temporal and spatial intermittencies within channel flow turbulence near transition. Phys. Rev. Fluids 2, 024603.10.1103/PhysRevFluids.2.024603
Landhal, M. T. 1980 A note on an algebraic instability of invscid parallel shear flow. J. Fluid Mech. 98, 243251.10.1017/S0022112080000122
Lee, M. J. & Moin, J. K. P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561583.10.1017/S0022112090000532
Limpert, E., Stahel, W. A. & Abbt, M. 2001 Log-normal distributions across the sciences: keys and clues. BioScience 51 (5), 341352.10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521539.10.1017/S0022112056000342
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.10.1017/S002211201000176X
Örlü, R. & Schlatter, P. 2011 On the fluctuating wall-shear stress in zero pressure-gradient turbulent boundary layer flows. Phys. Fluids 23 (2), 021704.10.1063/1.3555191
Panton, R. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37, 341383.10.1016/S0376-0421(01)00009-4
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430454.10.1017/jfm.2015.554
Pausch, M., Yang, Q., Hwang, Y. & Eckhardt, B. 2019 Quasilinear approximation for exact coherent states in parallel shear flows. Fluid Dyn. Res. 51, 011402.10.1088/1873-7005/aaadcc
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.10.1007/978-1-4757-6557-1
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), 2429.10.1073/pnas.1712645115
Rao, K. N., Narasimha, R. & Narayanan, M. A. B. 1971 The bursting phenomenon in a turbulent boundary layer. J. Fluid Mech. 48 (2), 339352.10.1017/S0022112071001605
Rawat, S., Cossu, C. & Rincon, F. 2014 Relative periodic orbits in plane Poiseuille flow. C. R. Méc. 342, 485489.10.1016/j.crme.2014.05.008
Rawat, S., Cossu, C. & Rincon, F. 2016 Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow. C. R. Méc. 344, 448455.10.1016/j.crme.2015.12.005
Rinaldi, E., Schlatter, P. & Bagheri, S. 2018 Edge state modulation by mean viscosity gradients. J. Fluid Mech. 838, 379403.10.1017/jfm.2017.921
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601639.10.1146/annurev.fl.23.010191.003125
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.10.1007/978-1-4613-0185-1
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, 037301.10.1103/PhysRevE.78.037301
Schneider, T. M., Marinc, D. & Eckhardt, B. 2010 Localized edge states nucleate turbulence in extended plane couette cells. J. Fluid Mech. 646, 441451.10.1017/S0022112009993144
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.10.1017/jfm.2013.286
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, 114.10.1098/rsta.2016.0089
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).10.1103/PhysRevE.93.021102
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, 2754.10.1017/S0022112083000634
Tailleur, J. & Kurchan, J. 2007 Probing rare physical trajectories with Lyapunov weighted dynamics. Nat. Phys. 3, 203207.10.1038/nphys515
Taylor, G. I. 1935 Turbulence in a contracting stream. Z. Angew. Math. Mech. 15, 9196.10.1002/zamm.19350150119
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 6776.10.1017/S0022112003003768
Toh, S. & Itano, T. 2005 Interaction between a large-scale structure and near-wall structures in channel flow. J. Fluid Mech. 524, 249262.10.1017/S002211200400237X
Virk, P., Mickley, H. & Smith, K. 1970 The ultimate asymptote and mean flow structure in Toms phenomenon. Trans. ASME E: J. Appl. Mech. 37, 488493.10.1115/1.3408532
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 41404143.10.1103/PhysRevLett.81.4140
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 15171534.10.1063/1.1566753
Waleffe, F. & Kim, J. 1997 How Streamwise Rolls and Streaks Self-Sustain in a Shear Flow (ed. Panton, R.), pp. 309332. Computational Mechanics Publications.
Wallace, J. M., Eckelmann, H. & Brodkey, R. S. 1972 The wall region in turbulent shear flow. J. Fluid Mech. 54 (1), 3948.10.1017/S0022112072000515
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.10.1103/PhysRevLett.98.204501
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, 213233.10.1017/S0022112008004618
Xi, L. & Bai, X. 2016 Marginal turbulent state of viscoelastic fluids: a polymer drag reduction perspective. Phys. Rev. E 93, 043118.
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, 028301.10.1103/PhysRevLett.108.028301
Zammert, S. & Eckhardt, B. 2014 Periodically bursting edge states in plane Poiseuille flow. Fluid. Dyn. Res. 46, 041419.10.1088/0169-5983/46/4/041419
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