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A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow

  • Brian F. Farrell (a1), Petros J. Ioannou (a2), Javier Jiménez (a3), Navid C. Constantinou (a4), Adrián Lozano-Durán (a3) and Marios-Andreas Nikolaidis (a2)...


The perspective of statistical state dynamics (SSD) has recently been applied to the study of mechanisms underlying turbulence in a variety of physical systems. An SSD is a dynamical system that evolves a representation of the statistical state of the system. An example of an SSD is the second-order cumulant closure referred to as stochastic structural stability theory (S3T), which has provided insight into the dynamics of wall turbulence, and specifically the emergence and maintenance of the roll/streak structure. S3T comprises a coupled set of equations for the streamwise mean and perturbation covariance, in which nonlinear interactions among the perturbations has been removed, restricting nonlinearity in the dynamics to that of the mean equation and the interaction between the mean and perturbation covariance. In this work, this quasi-linear restriction of the dynamics is used to study the structure and dynamics of turbulence in plane Poiseuille flow at moderately high Reynolds numbers in a closely related dynamical system, referred to as the restricted nonlinear (RNL) system. Simulations using this RNL system reveal that the essential features of wall-turbulence dynamics are retained. Consistent with previous analyses based on the S3T version of SSD, the RNL system spontaneously limits the support of its turbulence to a small set of streamwise Fourier components, giving rise to a naturally minimal representation of its turbulence dynamics. Although greatly simplified, this RNL turbulence exhibits natural-looking structures and statistics, albeit with quantitative differences from those in direct numerical simulations (DNS) of the full equations. Surprisingly, even when further truncation of the perturbation support to a single streamwise component is imposed, the RNL system continues to self-sustain turbulence with qualitatively realistic structure and dynamic properties. RNL turbulence at the Reynolds numbers studied is dominated by the roll/streak structure in the buffer layer and similar very large-scale structure (VLSM) in the outer layer. In this work, diagnostics of the structure, spectrum and energetics of RNL and DNS turbulence are used to demonstrate that the roll/streak dynamics supporting the turbulence in the buffer and logarithmic layer is essentially similar in RNL and DNS.


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Present address: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA.



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del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135144.
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329358.
Bakas, N. A. & Ioannou, P. J. 2013 Emergence of large scale structure in barotropic 𝛽-plane turbulence. Phys. Rev. Lett. 110, 224501.
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 32583269.
Bouchet, F., Nardini, C. & Tangarife, T. 2013 Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier–Stokes equations. J. Stat. Phys. 153 (4), 572625.
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.
Bullock, K. J., Cooper, R. E. & Abernathy, F. H. 1978 Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow. J. Fluid Mech. 88, 585608.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids 4, 16371650.
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2014a Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory. J. Atmos. Sci. 71 (5), 18181842.
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2016 Statistical state dynamics of jet–wave coexistence in barotropic beta-plane turbulence. J. Atmos. Sci. 73 (5), 22292253.
Constantinou, N. C., Lozano-Durán, A., Nikolaidis, M.-A., Farrell, B. F., Ioannou, P. J. & Jiménez, J. 2014b Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS. J. Phys.: Conf. Ser. 506, 012004.
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 7994.
Dallas, V., Vassilicos, J. C. & Hewitt, G. F. 2009 Stagnation point von Kármán coefficient. Phys. Rev. E 80, 046306.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 13901400.
Farrell, B. F. & Ioannou, P. J. 1996a Generalized stability. Part I. Autonomous operators. J. Atmos. Sci. 53, 20252040.
Farrell, B. F. & Ioannou, P. J. 1996b Generalized stability. Part II. Non-autonomous operators. J. Atmos. Sci. 53, 20412053.
Farrell, B. F. & Ioannou, P. J. 1999 Perturbation growth and structure in time dependent flows. J. Atmos. Sci. 56, 36223639.
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 21012118.
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 36523665.
Farrell, B. F. & Ioannou, P. J. 2008 Formation of jets by baroclinic turbulence. J. Atmos. Sci. 65, 33533375.
Farrell, B. F. & Ioannou, P. J. 2009a Emergence of jets from turbulence in the shallow-water equations on an equatorial beta plane. J. Atmos. Sci. 66, 31973207.
Farrell, B. F. & Ioannou, P. J. 2009b A stochastic structural stability theory model of the drift wave-zonal flow system. Phys. Plasmas 16, 112903.
Farrell, B. F. & Ioannou, P. J. 2009c A theory of baroclinic turbulence. J. Atmos. Sci. 66, 24442454.
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149196.
Farrell, B. F. & Ioannou, P. J. 2016 Structure and mechanism in a second-order statistical state dynamics model of self-sustaining turbulence in plane Couette flow. Phys. Rev. Fluids; (submitted, arXiv:1607.05020).
Farrell, B. F., Ioannou, P. J. & Nikolaidis, M.-A. 2016 Instability of the roll/streak structure induced by free-stream turbulence in pre-transitional Couette flow. Phys. Rev. Fluids; (submitted, arXiv:1607.05018).
Flores, O. & Jiménez, J. 2006 Effect of wall-boundary disturbances on turbulent channel flows. J. Fluid Mech. 566, 357376.
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.
Foias, C., Manley, O., Rosa, R. & Temam, R. 2001 Navier–Stokes Equations and Turbulence. Cambridge University Press.
Gayme, D. F.2010 A robust control approach to understanding nonlinear mechanisms in shear flow turbulence. PhD thesis, Caltech, Pasadena, CA, USA.
Gayme, D. F., McKeon, B. J., Papachristodoulou, A., Bamieh, B. & Doyle, J. C. 2010 A streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99119.
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.
Hama, F. R., Long, J. D. & Hegarty, J. C. 1957 On transition from laminar to turbulent flow. J. Appl. Phys. 28 (4), 388394.
Hamilton, K., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.
Hellström, L. H. O., Marusic, I. & Smits, A. J. 2016 Self-similarity of the large-scale motions in turbulent pipe flow. J. Fluid Mech. 792, R1.
Hellström, L. H. O., Sinha, A. & Smits, A. J. 2011 Visualizing the very-large-scale motions in turbulent pipe flow. Phys. Fluids 23, 011703.
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 128.
Jiménez, J. 1998 The largest scales of turbulent wall flows. In CTR Annual Research Briefs, pp. 137154. Stanford University.
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215236.
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335359.
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.
Kim, J. & Lim, J. 2000 A linear process in wall bounded turbulent shear flows. Phys. Fluids 12, 18851888.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 417422.
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741773.
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100130.
Lozano-Durán, A. & Jiménez, J. 2014a Effect of the computational domain on direct simulations of turbulent channels up to Re 𝜏 = 4200. Phys. Fluids 26, 011702.
Lozano-Durán, A. & Jiménez, J. 2014b Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432471.
Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65 (6), 19551966.
Marusic, I., McKeon, B. J., Monkewitz, P. A., Nagib, H. M. & Sreenivasan, K. R. 2010 Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids 22, 065103.
Millikan, C. B. 1938 A critical discussion of turbulent flows in channels and circular tubes. Proceedings of the Fifth International Congress for Applied Mechanics. Wiley.
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without walls. J. Fluid Mech. 723, 429455.
Nikolaidis, M.-A., Farrell, B. F., Ioannou, P. J., Gayme, D. F., Lozano-Durán, A. & Jiménez, J. 2016 A POD-based analysis of turbulence in the reduced nonlinear dynamics system. J. Phys.: Conf. Ser. 708, 012002.
Parker, J. B. & Krommes, J. A. 2013 Zonal flow as pattern formation. Phys. Plasmas 20, 100703.
Parker, J. B. & Krommes, J. A. 2014 Generation of zonal flows through symmetry breaking of statistical homogeneity. New J. Phys. 16 (3), 035006.
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, 515540.
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 16331656.
Thomas, V., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27, 105104.
Thomas, V., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26, 105112.
Toh, S. & Itano, T. 2005 Interaction between a large-scale structure and near-wall structures in channel flow. J. Fluid Mech. 524, 249262.
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.
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A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow

  • Brian F. Farrell (a1), Petros J. Ioannou (a2), Javier Jiménez (a3), Navid C. Constantinou (a4), Adrián Lozano-Durán (a3) and Marios-Andreas Nikolaidis (a2)...


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