Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-13T01:36:59.324Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  09 November 2016

Brian F. Farrell ,
Petros J. Ioannou ,
Javier Jiménez ,
Navid C. Constantinou Open the ORCID record for Navid C. Constantinou [Opens in a new window] ,
Adrián Lozano-Durán  and
Marios-Andreas Nikolaidis
Brian F. Farrell
Affiliation:
Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA
Petros J. Ioannou*
Affiliation:
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens, 157 84, Greece
Javier Jiménez
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, 28040, Madrid, Spain
Navid C. Constantinou
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 90293-0213, USA
Adrián Lozano-Durán
Affiliation:
School of Aeronautics, Universidad Politécnica de Madrid, 28040, Madrid, Spain
Marios-Andreas Nikolaidis
Affiliation:
Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens, 157 84, Greece
*
†Email address for correspondence: pjioannou@phys.uoa.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Boundary Layers: Boundary layer structure Instability: Parametric instability Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 290 - 315
Check for updates
Copyright
© 2016 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.)

Footnotes

‡

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

References

del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
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, 329–358.CrossRefGoogle Scholar
Bakas, N. A. & Ioannou, P. J. 2013 Emergence of large scale structure in barotropic 𝛽-plane turbulence. Phys. Rev. Lett. 110, 224501.CrossRefGoogle ScholarPubMed
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 3258–3269.CrossRefGoogle Scholar
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), 572–625.CrossRefGoogle 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.CrossRefGoogle Scholar
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, 585–608.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids 4, 1637–1650.CrossRefGoogle Scholar
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), 1818–1842.CrossRefGoogle Scholar
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), 2229–2253.CrossRefGoogle Scholar
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.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
Dallas, V., Vassilicos, J. C. & Hewitt, G. F. 2009 Stagnation point von Kármán coefficient. Phys. Rev. E 80, 046306.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390–1400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996a Generalized stability. Part I. Autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996b Generalized stability. Part II. Non-autonomous operators. J. Atmos. Sci. 53, 2041–2053.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1999 Perturbation growth and structure in time dependent flows. J. Atmos. Sci. 56, 3622–3639.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 3652–3665.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2008 Formation of jets by baroclinic turbulence. J. Atmos. Sci. 65, 3353–3375.CrossRefGoogle Scholar
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, 3197–3207.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2009b A stochastic structural stability theory model of the drift wave-zonal flow system. Phys. Plasmas 16, 112903.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2009c A theory of baroclinic turbulence. J. Atmos. Sci. 66, 2444–2454.CrossRefGoogle 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.CrossRefGoogle Scholar
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).Google Scholar
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).Google Scholar
Flores, O. & Jiménez, J. 2006 Effect of wall-boundary disturbances on turbulent channel flows. J. Fluid Mech. 566, 357–376.CrossRefGoogle Scholar
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.CrossRefGoogle Scholar
Foias, C., Manley, O., Rosa, R. & Temam, R. 2001 Navier–Stokes Equations and Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Gayme, D. F.2010 A robust control approach to understanding nonlinear mechanisms in shear flow turbulence. PhD thesis, Caltech, Pasadena, CA, USA.Google Scholar
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, 99–119.CrossRefGoogle 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.CrossRefGoogle Scholar
Hama, F. R., Long, J. D. & Hegarty, J. C. 1957 On transition from laminar to turbulent flow. J. Appl. Phys. 28 (4), 388–394.CrossRefGoogle Scholar
Hamilton, K., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hellström, L. H. O., Sinha, A. & Smits, A. J. 2011 Visualizing the very-large-scale motions in turbulent pipe flow. Phys. Fluids 23, 011703.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
Jiménez, J. 1998 The largest scales of turbulent wall flows. In CTR Annual Research Briefs, pp. 137–154. Stanford University.Google Scholar
Jiménez, J. 2013 Near-wall turbulence. Phys. Fluids 25 (10), 101302.CrossRefGoogle Scholar
Jiménez, J. & Hoyas, S. 2008 Turbulent fluctuations above the buffer layer of wall-bounded flows. J. Fluid Mech. 611, 215–236.CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kim, J. & Lim, J. 2000 A linear process in wall bounded turbulent shear flows. Phys. Fluids 12, 1885–1888.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 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
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100–130.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Lozano-Durán, A. & Jiménez, J. 2014b Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades. J. Fluid Mech. 759, 432–471.CrossRefGoogle Scholar
Marston, J. B., Conover, E. & Schneider, T. 2008 Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65 (6), 1955–1966.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Mizuno, Y. & Jiménez, J. 2013 Wall turbulence without walls. J. Fluid Mech. 723, 429–455.CrossRefGoogle Scholar
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.Google Scholar
Parker, J. B. & Krommes, J. A. 2013 Zonal flow as pattern formation. Phys. Plasmas 20, 100703.CrossRefGoogle Scholar
Parker, J. B. & Krommes, J. A. 2014 Generation of zonal flows through symmetry breaking of statistical homogeneity. New J. Phys. 16 (3), 035006.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
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Srinivasan, K. & Young, W. R. 2012 Zonostrophic instability. J. Atmos. Sci. 69 (5), 1633–1656.CrossRefGoogle Scholar
Thomas, V., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27, 105104.CrossRefGoogle Scholar
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.CrossRefGoogle 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.CrossRefGoogle Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow, 2nd edn. Cambridge University Press.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883–900.CrossRefGoogle Scholar