×

zbMATH — the first resource for mathematics

Restricted nonlinear model for high- and low-drag events in plane channel flow. (English) Zbl 1415.76320
Summary: 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 S. Toh and T. Itano [ibid. 481, 67–76 (2003; Zbl 1034.76014)]. 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 [L. Xi and M. D. Graham, “Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote”, Phys. Rev. Lett. 108, No. 2, Article ID 028301, 5 p. (2012; doi:10.1103/PhysRevLett.108.028301)]. 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.

MSC:
76F40 Turbulent boundary layers
76F06 Transition to turbulence
76F10 Shear flows and turbulence
Software:
channelflow
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abe, H.; Antonia, R. A.; Toh, S., Large-scale structures in a turbulent channel flow with a minimal streamwise flow unit, J. Fluid Mech., 850, 733-768, (2018) · Zbl 1415.76319
[2] Alfredsson, P. H.; Örlü, R.; Schlatter, P., The viscous sublayer revisited - exploiting self-similarity to determine the wall position and friction velocity, Exp. Fluids, 51, 1, 271-280, (2011)
[3] Alizard, F., Linear stability of optimal streaks in the log-layer of turbulent channel flows, Phys. Fluids, 27, (2015)
[4] Alizard, F., Invariant solutions in a channel flow using a minimal restricted nonlinear model, C. R. Méc., 345, 117-124, (2017)
[5] Biau, D.; Bottaro, A., An optimal path to transition in a duct, Phil. Trans. R. Soc. Lond. A, 367, 529-544, (2009) · Zbl 1221.76094
[6] Blackburn, H. M.; Hall, P.; Sherwin, S. J., Lower branch equilibria in couette flow: the emergence of canonical states for arbitrary shear flows, J. Fluid Mech., 726, R2, (2013) · Zbl 1287.76098
[7] Bretheim, J. U.; Meneveau, C.; Gayme, D. F., Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel, Phys. Fluids, 27, (2015)
[8] Bretheim, J. U.; Meneveau, C.; Gayme, D. F., A restricted nonlinear large eddy simulation model for high Reynolds number flows, J. Turbul., 19, 141-166, (2018)
[9] Butler, K. M.; Farrell, B. F., Optimal perturbations and streak spacing in wall-bounded turbulent shear flow, Phys. Fluids, 5, 774-777, (1993)
[10] Cossu, C.; Hwang, Y., Self-sustaining processes at all scales in wall-bounded turbulent shear flows, Phil. Trans. R. Soc. Lond. A, 375, 1-14, (2017)
[11] Curry, J. H.; Herring, J. R.; Loncaric, J.; Orszag, S. A., Order and disorder in two- and three-dimensional Bénard convection, J. Fluid Mech., 147, 1-38, (1984) · Zbl 0547.76093
[12] Diwan, S. S.; Morrison, J. F., Spectral structure and linear mechanisms in a rapidly distorted boundary layer, Intl J. Heat Fluid Flow, 67, (2017)
[13] Duguet, Y.; Willis, A. P.; Kerswell, R. R., Transition in pipe flow: the saddle structure on the boundary of turbulence, J. Fluid Mech., 613, 255-274, (2008) · Zbl 1151.76495
[14] Eckhardt, B., Doubly localized states in plane Couette flow, J. Fluid Mech., 758, 1-4, (2014)
[15] Farrell, B. F.; Gayme, D. F.; Ioannou, P. J., A statistical state dynamics approach to wall turbulence, Phil. Trans. R. Soc. Lond. A, 375, 2089, (2017)
[16] Farrell, B. F.; Ioannou, P. J., Generalized stability theory. Part II: nonautonomous operators, J. Atmos. Sci., 53, 14, 2041-2053, (1996)
[17] Farrell, B. F.; Ioannou, P. J., Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow, J. Fluid Mech., 708, 149-196, (2012) · Zbl 1275.76125
[18] Farrell, B. F.; Ioannou, P. J.; Jiménez, J.; Constantinou, N. C.; Lozano-Durán, A.; Nikolaidis, M. A., 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, (2016) · Zbl 1383.76225
[19] Gibson, J. F.; Brand, E., Spanwise-localized solutions of planar shear flows, J. Fluid Mech., 745, 25-61, (2014)
[20] Gibson, J. F.; Halcrow, J.; Cvitanovic, P., Visualizing the geometry of state space in plane Couette flow, J. Fluid Mech., 611, 107-130, (2008) · Zbl 1151.76453
[21] Hall, P.; Sherwin, S., Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures, J. Fluid Mech., 661, 178-205, (2010) · Zbl 1205.76085
[22] Hamilton, J. M.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032
[23] Hunt, J. C. R.; Carruthers, D. J., Rapid distortion theory and the problems of turbulence, J. Fluid Mech., 212, 497-532, (1990) · Zbl 0692.76054
[24] Hwang, Y.; Cossu, C., Linear non-normal energy amplification of harmonic and stochastic forcing in turbulent channel flow, J. Fluid Mech., 664, 51-73, (2010) · Zbl 1221.76104
[25] Hwang, Y.; Willis, A. P.; Cossu, C., Invariant solutions of minimal large-scale structures in turbulent channel flow for Re_𝜏 up to 1000, J. Fluid Mech., 802, R1, (2016)
[26] Itano, T.; Toh, S., The dynamics of bursting process in wall turbulence, J. Phys. Soc. Japan, 70, 703-716, (2001)
[27] Jiménez, J., Coherent structures in wall-bounded turbulence, J. Fluid Mech., 842, P1, (2018)
[28] Jiménez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, 213-240, (1991) · Zbl 0721.76040
[29] Jiménez, J.; Pinelli, A., The autonomous cycle of near-wall turbulence, J. Fluid Mech., 389, 335-359, (1999) · Zbl 0948.76025
[30] Kawahara, G.; Uhlmann, M.; Veen, L. V., The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech., 44, 203-225, (2012) · Zbl 1352.76031
[31] Khapko, T.; Kreilos, T.; Schlatter, P.; Duguet, Y.; Eckardt, B.; Henningson, D. S., Localized edge states in the asymptotic suction boundary layer, J. Fluid Mech., 717, 1-11, (2013) · Zbl 1284.76106
[32] Kim, H. T.; Kline, S. J.; Reynolds, W. C., The production of turbulence near a smooth wall in a turbulent boundary layer, J. Fluid Mech., 50, 1, 133-160, (1971)
[33] Kline, S. J.; Reynolds, W. C.; Schraub, F. A.; Runstadler, P. W., The structure of turbulent boundary layers, J. Fluid Mech., 30, 4, 741-773, (1967)
[34] Kreilos, T.; Veble, G.; Schneider, T. M.; Eckhardt, B., Edge states for the turbulence transition in the asymptotic suction boundary layer, J. Fluid Mech., 726, 100-122, (2013) · Zbl 1287.76122
[35] Kushwaha, A.; Park, J. S.; Graham, M. D., Temporal and spatial intermittencies within channel flow turbulence near transition, Phys. Rev. Fluids, 2, (2017)
[36] Landhal, M. T., A note on an algebraic instability of invscid parallel shear flow, J. Fluid Mech., 98, 243-251, (1980) · Zbl 0428.76049
[37] Lee, M. J.; Moin, J. K. P., Structure of turbulence at high shear rate, J. Fluid Mech., 216, 561-583, (1990)
[38] Limpert, E.; Stahel, W. A.; Abbt, M., Log-normal distributions across the sciences: keys and clues, BioScience, 51, 5, 341-352, (2001)
[39] Malkus, W. V. R., Outline of a theory of turbulent shear flow, J. Fluid Mech., 1, 521-539, (1956) · Zbl 0073.20803
[40] Mckeon, B. J.; Sharma, A. S., A critical-layer framework for turbulent pipe flow, J. Fluid Mech., 658, 336-382, (2010) · Zbl 1205.76138
[41] Örlü, R.; Schlatter, P., On the fluctuating wall-shear stress in zero pressure-gradient turbulent boundary layer flows, Phys. Fluids, 23, 2, (2011)
[42] Panton, R., Overview of the self-sustaining mechanisms of wall turbulence, Prog. Aerosp. Sci., 37, 341-383, (2001)
[43] Park, J. S.; Graham, M. D., Exact coherent states and connections to turbulent dynamics in minimal channel flow, J. Fluid Mech., 782, 430-454, (2015) · Zbl 1381.76097
[44] Pausch, M.; Yang, Q.; Hwang, Y.; Eckhardt, B., Quasilinear approximation for exact coherent states in parallel shear flows, Fluid Dyn. Res., 51, (2019)
[45] Peyret, R., Spectral Methods for Incompressible Viscous Flow, (2002), Springer · Zbl 1005.76001
[46] Ragone, F.; Wouters, J.; Bouchet, F., Computation of extreme heat waves in climate models using a large deviation algorithm, Proc. Natl Acad. Sci., 115, 1, 24-29, (2017)
[47] Rao, K. N.; Narasimha, R.; Narayanan, M. A. B., The bursting phenomenon in a turbulent boundary layer, J. Fluid Mech., 48, 2, 339-352, (1971)
[48] Rawat, S.; Cossu, C.; Rincon, F., Relative periodic orbits in plane Poiseuille flow, C. R. Méc., 342, 485-489, (2014)
[49] Rawat, S.; Cossu, C.; Rincon, F., Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow, C. R. Méc., 344, 448-455, (2016)
[50] Rinaldi, E.; Schlatter, P.; Bagheri, S., Edge state modulation by mean viscosity gradients, J. Fluid Mech., 838, 379-403, (2018)
[51] Robinson, S. K., Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech., 23, 1, 601-639, (1991)
[52] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, 142, (2001), Springer · Zbl 0966.76003
[53] Schneider, T. M.; Gisbon, J. F.; Lagha, M.; Lillo, F. D.; Eckhardt, B., Laminar-turbulent boundary in plane Couette flow, Phys. Rev. E, 78, (2008)
[54] Schneider, T. M.; Marinc, D.; Eckhardt, B., Localized edge states nucleate turbulence in extended plane couette cells, J. Fluid Mech., 646, 441-451, (2010) · Zbl 1189.76258
[55] Sharma, A. S.; Mckeon, B. J., On coherent structure in wall turbulence, J. Fluid Mech., 728, 196-238, (2013) · Zbl 1291.76173
[56] Sharma, A. S.; Moarref, R.; Mckeon, B. J., Scaling and interaction of self-similar modes in models of high Reynolds number wall turbulence, Phil. Trans. R. Soc. Lond. A, 375, 1-14, (2017) · Zbl 1404.76129
[57] Sharma, A. S.; Moarref, R.; Mckeon, B. J.; Park, J. S.; Graham, M. D.; Willis, A. P., Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall-turbulence, Phys. Rev. E, 93, (2016)
[58] Smith, C. R.; Metzler, S. P., The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer, J. Fluid Mech., 129, 27-54, (1983)
[59] Tailleur, J.; Kurchan, J., Probing rare physical trajectories with Lyapunov weighted dynamics, Nat. Phys., 3, 203-207, (2007)
[60] Taylor, G. I., Turbulence in a contracting stream, Z. Angew. Math. Mech., 15, 91-96, (1935) · JFM 61.0927.01
[61] Toh, S.; Itano, T., A periodic-like solution in channel flow, J. Fluid Mech., 481, 67-76, (2003) · Zbl 1034.76014
[62] Toh, S.; Itano, T., Interaction between a large-scale structure and near-wall structures in channel flow, J. Fluid Mech., 524, 249-262, (2005) · Zbl 1065.76553
[63] Virk, P.; Mickley, H.; Smith, K., The ultimate asymptote and mean flow structure in Toms phenomenon, Trans. ASME E: J. Appl. Mech., 37, 488-493, (1970)
[64] Waleffe, F., Three-dimensional coherent states in plane shear flows, Phys. Rev. Lett., 81, 4140-4143, (1998)
[65] Waleffe, F., Homotopy of exact coherent structures in plane shear flows, Phys. Fluids, 15, 1517-1534, (2003) · Zbl 1186.76556
[66] Waleffe, F.; Kim, J.; Panton, R., How Streamwise Rolls and Streaks Self-Sustain in a Shear Flow, 309-332, (1997), Computational Mechanics Publications
[67] Wallace, J. M.; Eckelmann, H.; Brodkey, R. S., The wall region in turbulent shear flow, J. Fluid Mech., 54, 1, 39-48, (1972)
[68] Wang, J.; Gibson, J.; Waleffe, F., Lower branch coherent states in shear flows: transition and control, Phys. Rev. Lett., 98, (2007)
[69] Willis, A. P.; Kerswell, R. R., Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarization and localized ‘edge’ states, J. Fluid Mech., 619, 213-233, (2009) · Zbl 1156.76395
[70] Xi, L.; Bai, X., Marginal turbulent state of viscoelastic fluids: a polymer drag reduction perspective, Phys. Rev. E, 93, (2016)
[71] Xi, L.; Graham, M. D., Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote, Phys. Rev. Lett., 108, (2012)
[72] Zammert, S.; Eckhardt, B., Periodically bursting edge states in plane Poiseuille flow, Fluid. Dyn. Res., 46, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.