# zbMATH — the first resource for mathematics

A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. (English) Zbl 1383.76225
Summary: 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.

##### MSC:
 76F40 Turbulent boundary layers 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76F65 Direct numerical and large eddy simulation of turbulence 76E99 Hydrodynamic stability
Full Text:
##### References:
 [1] Del Álamo, J. C.; Jiménez, J.; Zandonade, P.; Moser, R. D., Scaling of the energy spectra of turbulent channels, J. Fluid Mech., 500, 135-144, (2004) · Zbl 1059.76031 [2] Del Álamo, J. C.; Jiménez, J.; Zandonade, P.; Moser, R. D., Self-similar vortex clusters in the turbulent logarithmic region, J. Fluid Mech., 561, 329-358, (2006) · Zbl 1157.76346 [3] Bakas, N. A.; Ioannou, P. J., Emergence of large scale structure in barotropic 𝛽-plane turbulence, Phys. Rev. Lett., 110, (2013) [4] Bamieh, B.; Dahleh, M., Energy amplification in channel flows with stochastic excitation, Phys. Fluids, 13, 3258-3269, (2001) · Zbl 1184.76042 [5] Bouchet, F.; Nardini, C.; Tangarife, T., Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations, J. Stat. Phys., 153, 4, 572-625, (2013) · Zbl 1292.82031 [6] 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) [7] Bullock, K. J.; Cooper, R. E.; Abernathy, F. H., Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow, J. Fluid Mech., 88, 585-608, (1978) [8] Butler, K. M.; Farrell, B. F., Three-dimensional optimal perturbations in viscous shear flows, Phys. Fluids, 4, 1637-1650, (1992) [9] Constantinou, N. C.; Farrell, B. F.; Ioannou, P. J., Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory, J. Atmos. Sci., 71, 5, 1818-1842, (2014) [10] Constantinou, N. C.; Farrell, B. F.; Ioannou, P. J., Statistical state dynamics of jet – wave coexistence in barotropic beta-plane turbulence, J. Atmos. Sci., 73, 5, 2229-2253, (2016) [11] Constantinou, N. C.; Lozano-Durán, A.; Nikolaidis, M.-A.; Farrell, B. F.; Ioannou, P. J.; Jiménez, J., Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS, J. Phys.: Conf. Ser., 506, (2014) [12] Cossu, C.; Pujals, G.; Depardon, S., Optimal transient growth and very large-scale structures in turbulent boundary layers, J. Fluid Mech., 619, 79-94, (2009) · Zbl 1156.76400 [13] Dallas, V.; Vassilicos, J. C.; Hewitt, G. F., Stagnation point von Kármán coefficient, Phys. Rev. E, 80, (2009) [14] Drazin, P. G.; Reid, W. H., Hydrodynamic Stability, (1981), Cambridge University Press [15] Farrell, B. F.; Ioannou, P. J., Optimal excitation of three-dimensional perturbations in viscous constant shear flow, Phys. Fluids A, 5, 1390-1400, (1993) · Zbl 0779.76030 [16] Farrell, B. F.; Ioannou, P. J., Generalized stability. Part I. Autonomous operators, J. Atmos. Sci., 53, 2025-2040, (1996) [17] Farrell, B. F.; Ioannou, P. J., Generalized stability. Part II. Non-autonomous operators, J. Atmos. Sci., 53, 2041-2053, (1996) [18] Farrell, B. F.; Ioannou, P. J., Perturbation growth and structure in time dependent flows, J. Atmos. Sci., 56, 3622-3639, (1999) [19] Farrell, B. F.; Ioannou, P. J., Structural stability of turbulent jets, J. Atmos. Sci., 60, 2101-2118, (2003) [20] Farrell, B. F.; Ioannou, P. J., Structure and spacing of jets in barotropic turbulence, J. Atmos. Sci., 64, 3652-3665, (2007) · Zbl 1127.90042 [21] Farrell, B. F.; Ioannou, P. J., Formation of jets by baroclinic turbulence, J. Atmos. Sci., 65, 3353-3375, (2008) [22] Farrell, B. F.; Ioannou, P. J., Emergence of jets from turbulence in the shallow-water equations on an equatorial beta plane, J. Atmos. Sci., 66, 3197-3207, (2009) [23] Farrell, B. F.; Ioannou, P. J., A stochastic structural stability theory model of the drift wave-zonal flow system, Phys. Plasmas, 16, (2009) [24] Farrell, B. F.; Ioannou, P. J., A theory of baroclinic turbulence, J. Atmos. Sci., 66, 2444-2454, (2009) [25] 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 [26] Farrell, B. F. & Ioannou, P. J.2016Structure 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). [27] Farrell, B. F., Ioannou, P. J. & Nikolaidis, M.-A.2016Instability of the roll/streak structure induced by free-stream turbulence in pre-transitional Couette flow. Phys. Rev. Fluids; (submitted, arXiv:1607.05018). [28] Flores, O.; Jiménez, J., Effect of wall-boundary disturbances on turbulent channel flows, J. Fluid Mech., 566, 357-376, (2006) · Zbl 1275.76146 [29] Flores, O.; Jiménez, J., Hierarchy of minimal flow units in the logarithmic layer, Phys. Fluids, 22, (2010) [30] Foias, C.; Manley, O.; Rosa, R.; Temam, R., Navier-Stokes Equations and Turbulence, (2001), Cambridge University Press · Zbl 0994.35002 [31] Gayme, D. F.2010 A robust control approach to understanding nonlinear mechanisms in shear flow turbulence. PhD thesis, Caltech, Pasadena, CA, USA. [32] Gayme, D. F.; Mckeon, B. J.; Papachristodoulou, A.; Bamieh, B.; Doyle, J. C., A streamwise constant model of turbulence in plane Couette flow, J. Fluid Mech., 665, 99-119, (2010) · Zbl 1225.76149 [33] 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 [34] Hama, F. R.; Long, J. D.; Hegarty, J. C., On transition from laminar to turbulent flow, J. Appl. Phys., 28, 4, 388-394, (1957) [35] Hamilton, K.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032 [36] Hellström, L. H. O.; Marusic, I.; Smits, A. J., Self-similarity of the large-scale motions in turbulent pipe flow, J. Fluid Mech., 792, R1, (2016) · Zbl 1381.76119 [37] Hellström, L. H. O.; Sinha, A.; Smits, A. J., Visualizing the very-large-scale motions in turbulent pipe flow, Phys. Fluids, 23, (2011) [38] Hutchins, N.; Marusic, I., Evidence of very long meandering features in the logarithmic region of turbulent boundary layers, J. Fluid Mech., 579, 1-28, (2007) · Zbl 1113.76004 [39] Jiménez, J.1998The largest scales of turbulent wall flows. In CTR Annual Research Briefs, pp. 137-154. Stanford University. [40] Jiménez, J., Near-wall turbulence, Phys. Fluids, 25, 10, (2013) [41] Jiménez, J.; Hoyas, S., Turbulent fluctuations above the buffer layer of wall-bounded flows, J. Fluid Mech., 611, 215-236, (2008) · Zbl 1151.76512 [42] Jiménez, J.; Pinelli, A., The autonomous cycle of near-wall turbulence, J. Fluid Mech., 389, 335-359, (1999) · Zbl 0948.76025 [43] Jovanović, M. R.; Bamieh, B., Componentwise energy amplification in channel flows, J. Fluid Mech., 534, 145-183, (2005) · Zbl 1074.76016 [44] Kim, J.; Lim, J., A linear process in wall bounded turbulent shear flows, Phys. Fluids, 12, 1885-1888, (2000) · Zbl 1184.76284 [45] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071 [46] Kim, K. C.; Adrian, R. J., Very large-scale motion in the outer layer, Phys. Fluids, 11, 417-422, (1999) · Zbl 1147.76430 [47] Kline, S. J.; Reynolds, W. C.; Schraub, F. A.; Runstadler, P. W., The structure of turbulent boundary layers, J. Fluid Mech., 30, 741-773, (1967) [48] Lozano-Durán, A.; Flores, O.; Jiménez, J., The three-dimensional structure of momentum transfer in turbulent channels, J. Fluid Mech., 694, 100-130, (2012) · Zbl 1250.76108 [49] Lozano-Durán, A.; Jiménez, J., Effect of the computational domain on direct simulations of turbulent channels up to Re_𝜏 = 4200, Phys. Fluids, 26, (2014) [50] Lozano-Durán, A.; Jiménez, J., Time-resolved evolution of coherent structures in turbulent channels: characterization of eddies and cascades, J. Fluid Mech., 759, 432-471, (2014) [51] Marston, J. B.; Conover, E.; Schneider, T., Statistics of an unstable barotropic jet from a cumulant expansion, J. Atmos. Sci., 65, 6, 1955-1966, (2008) [52] Marusic, I.; Mckeon, B. J.; Monkewitz, P. A.; Nagib, H. M.; Sreenivasan, K. R., Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues, Phys. Fluids, 22, (2010) · Zbl 1190.76086 [53] Millikan, C. B., A critical discussion of turbulent flows in channels and circular tubes, Proceedings of the Fifth International Congress for Applied Mechanics, (1938), Wiley · JFM 65.0993.01 [54] Mizuno, Y.; Jiménez, J., Wall turbulence without walls, J. Fluid Mech., 723, 429-455, (2013) · Zbl 1287.76137 [55] Nikolaidis, M.-A.; Farrell, B. F.; Ioannou, P. J.; Gayme, D. F.; Lozano-Durán, A.; Jiménez, J., A POD-based analysis of turbulence in the reduced nonlinear dynamics system, J. Phys.: Conf. Ser., 708, (2016) [56] Parker, J. B.; Krommes, J. A., Zonal flow as pattern formation, Phys. Plasmas, 20, (2013) [57] Parker, J. B.; Krommes, J. A., Generation of zonal flows through symmetry breaking of statistical homogeneity, New J. Phys., 16, 3, (2014) [58] Rawat, S.; Cossu, C.; Hwang, Y.; Rincon, F., On the self-sustained nature of large-scale motions in turbulent Couette flow, J. Fluid Mech., 782, 515-540, (2015) · Zbl 1381.76098 [59] Reddy, S. C.; Henningson, D. S., Energy growth in viscous channel flows, J. Fluid Mech., 252, 209-238, (1993) · Zbl 0789.76026 [60] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, (2001), Springer · Zbl 0966.76003 [61] Sharma, A. S.; Mckeon, B. J., On coherent structure in wall turbulence, J. Fluid Mech., 728, 196-238, (2013) · Zbl 1291.76173 [62] Srinivasan, K.; Young, W. R., Zonostrophic instability, J. Atmos. Sci., 69, 5, 1633-1656, (2012) [63] Thomas, V.; Farrell, B. F.; Ioannou, P. J.; Gayme, D. F., A minimal model of self-sustaining turbulence, Phys. Fluids, 27, (2015) [64] Thomas, V.; Lieu, B. K.; Jovanović, M. R.; Farrell, B. F.; Ioannou, P. J.; Gayme, D. F., Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow, Phys. Fluids, 26, (2014) [65] 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 [66] Townsend, A. A., The Structure of Turbulent Shear Flow, (1976), Cambridge University Press · Zbl 0325.76063 [67] Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 883-900, (1997)
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.