The quadrant analysis of the intense tangential Reynolds stress in plane turbulent channels is generalized to three-dimensional structures (Qs), with special emphasis on the logarithmic and outer layers. Wall-detached Qs are background stress fluctuations. They are small and isotropically oriented, and their contributions to the mean stress cancel. Wall-attached Qs are larger, and carry most of the mean Reynolds stresses. They form a family of roughly self-similar objects that become increasingly complex away from the wall, resembling the vortex clusters in del Álamo et al. (J. Fluid Mech., vol. 561, 2006, pp. 329–358). Individual Qs have fractal dimensions of the order of $D= 2$, slightly fuller than the clusters. They can be described as ‘sponges of flakes’, while vortex clusters are ‘sponges of strings’. The number of attached Qs decays away from the wall, but the fraction of the stress that they carry is independent of their sizes. A substantial fraction of the stress resides in a few large objects extending beyond the centreline, reminiscent of the very large structures of several authors. The predominant logarithmic-layer structure is a side-by-side pair of a sweep (Q4) and an ejection (Q2), with an associated cluster, and shares dimensions and stresses with the conjectured attached eddies of Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120). Those attached eddies tend to be aligned streamwise from each other, located near the side walls between the low- and high-velocity large-scale streaks, but that organization does not extend far enough to explain the very long structures in the centre of the channel.

The effect of domain size on direct numerical simulations of turbulent channels with periodic boundary conditions is studied. New simulations are presented up to Reτ = 4179 in boxes with streamwise and spanwise sizes of 2πh × πh, where h is the channel half-height. It is found that this domain is large enough to reproduce the one-point statistics of larger boxes. A simulation in a box of size 60πh × 6πh is used to show that a contour of the two-dimensional premultiplied spectrum of the streamwise velocity containing 80% of the kinetic energy closes at λx ≈ 100h.

A novel approach to the study of the kinematics and dynamics of turbulent flows is presented. The method involves tracking in time coherent structures, and provides all of the information required to characterize eddies from birth to death. Spatially and temporally well-resolved DNSs of channel data at $\mathit{Re}_{{\it\tau}}=930{-}4200$ are used to analyse the evolution of three-dimensional sweeps, ejections (Lozano-Durán et al., J. Fluid Mech., vol. 694, 2012, pp. 100–130) and clusters of vortices (del Álamo et al., J. Fluid Mech., vol. 561, 2006, pp. 329–358). The results show that most of the eddies remain small and do not last for long times, but that some become large, attach to the wall and extend across the logarithmic layer. The latter are geometrically and temporally self-similar, with lifetimes proportional to their size (or distance from the wall), and their dynamics is controlled by the mean shear near their centre of gravity. They are responsible for most of the total momentum transfer. Their origin, eventual disappearance, and history are investigated and characterized, including their advection velocity at different wall distances and the temporal evolution of their size. Reinforcing previous results, the symmetry found between sweeps and ejections supports the idea that they are not independent structures, but different manifestations of larger quasi-streamwise rollers in which they are embedded. Spatially localized direct and inverse cascades are respectively associated with the splitting and merging of individual structures, as in the models of Richardson (Proc. R. Soc. Lond. A, vol. 97(686), 1920, pp. 354–373) or Obukhov (Izv. Akad. Nauk USSR, Ser. Geogr. Geofiz., vol. 5(4), 1941, pp. 453–466). It is found that the direct cascade predominates, but that both directions are roughly comparable. Most of the merged or split fragments have sizes of the order of a few Kolmogorov viscous units, but a substantial fraction of the growth and decay of the larger eddies is due to a self-similar inertial process in which eddies merge and split in fragments spanning a wide range of scales.

We develop a method to estimate space-time flow statistics from a limited set of known data. While previous work has focused on modeling spatial or temporal statistics independently, space-time statistics carry fundamental information about the physics and coherent motions of the flow and provide a starting point for low-order modeling and flow control efforts. The method is derived using a statistical interpretation of resolvent analysis. The central idea of our approach is to use known data to infer the statistics of the nonlinear terms that constitute a forcing on the linearized Navier-Stokes equations, which in turn imply values for the remaining unknown flow statistics through application of the resolvent operator. Rather than making an a priori rank-1 assumption, our method allows the known input data to select the most relevant portions of the resolvent operator for describing the data, making it well-suited for highrank turbulent flows. We demonstrate the predictive capabilities of the method using two examples: the Ginzburg-Landau equation, which serves as a convenient model for a convectively unstable flow, and a turbulent channel flow at low Reynolds number. * Email address for correspondence: towne@umich.edu power spectral densities (PSDs) using knowledge of the mean flow field and power spectra at a few locations. This is accomplished using a least-squares fit at each frequency between the known power spectra and the leading singular response mode obtained from the resolvent operator (McKeon & Sharma 2010), which is derived from the linearized Navier-Stokes equations. This strategy explicitly assumes that the spectral content at frequencies of interest is dominated by the leading resolvent mode, and the method performs well when the matching points are located in regions where this hypothesis is valid. Specifically, excellent PSD estimates were obtained for the flow over a backward-facing step (Beneddine et al. 2016) and an initially laminar jet (Beneddine et al. 2017).Zare et al. (2017) developed a method that uses arbitrary known entries in the spatial covariance tensor to estimate the remaining unknown entries. Their approach is also based on linearized flow equations and entails solving a convex optimization problem that determines a matrix controlling the structure and statistics of the associated nonlinear forcing terms. The optimization problem is subject to two constraints on the estimated covariance tensor: it must reproduce the known entries and obey a Lyapunov equation that relates the forcing and flow statistics. The constrained optimization problem is computationally demanding and requires a customized algorithm (Zare et al. 2015(Zare et al. , 2017.The objective of the present paper is to build on these previous methods to estimate unknown two-point space-time flow statistics. Both the PSDs (one-point temporal statistics) and spatial covariances (two-point spatial statistics) are subsets of two-point space-time correlations, so our approach represents a generalization of these previous m...

A. (2016) 'A statistical state dynamics-based study of the structure and mechanism of largescale motions in plane Poiseuille flow', 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 non-linear (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. † Email address for correspondence: pjioannou@phys.uoa.gr arXiv:1512.06018v4 [physics.flu-dyn]

Two observations drawn from a thoroughly validated direct numerical simulation of the canonical spatially developing, zero-pressure gradient, smooth, flat-plate boundary layer are presented here. The first is that, for bypass transition in the narrow sense defined herein, we found that the transitional-turbulent spot inception mechanism is analogous to the secondary instability of boundary-layer natural transition, namely a spanwise vortex filament becomes a Λ vortex and then, a hairpin packet. Long streak meandering does occur but usually when a streak is infected by a nearby existing transitionalturbulent spot. Streak waviness and breakdown are, therefore, not the mechanisms for the inception of transitional-turbulent spots found here. Rather, they only facilitate the growth and spreading of existing transitional-turbulent spots. The second observation is the discovery, in the inner layer of the developed turbulent boundary layer, of what we call turbulent-turbulent spots. These turbulent-turbulent spots are dense concentrations of small-scale vortices with high swirling strength originating from hairpin packets. Although structurally quite similar to the transitional-turbulent spots, these turbulent-turbulent spots are generated locally in the fully turbulent environment, and they are persistent with a systematic variation of detection threshold level. They exert indentation, segmentation, and termination on the viscous sublayer streaks, and they coincide with local concentrations of high levels of Reynolds shear stress, enstrophy, and temperature fluctuations. The sublayer streaks seem to be passive and are often simply the rims of the indentation pockets arising from the turbulent-turbulent spots.boundary layer | transition | turbulence | direct numerical simulation T he zero-pressure gradient, smooth, flat-plate boundary layer (ZPGSFPBL) is the simplest viscous external flow. It serves as the idealized limiting case and calibration benchmark of atmospheric and oceanic planetary boundary layers as well as aeronautical, maritime, and automotive boundary-layer flows. For over 60 y since the work by Theodorsen (1), a central theme in fundamental fluid mechanics research has been the search for the constitutive coherent structure in the turbulent ZPGSFPBL, particularly inside the near-wall/inner layer less than ∼100 viscous units away from the plate where the production and dissipation of turbulence kinetic energy reach their peaks (2-6). When the nature of the inner-layer structure and dynamics is thoroughly understood, this understanding can be incorporated in turbulence theory and predictive modeling.Decades of research have produced an apparent consensus view (7-9) that the inner layer, which consists of the buffer layer and the viscous sublayer, of a turbulent ZPGSFPBL is populated by randomly distributed quasistreamwise vortices as well as elongated high-and low-momentum streaks. Streaks are thought to actively participate in a self-sustaining bursting cycle that includes streak generation, lift up, oscil...

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Non-equilibrium wall turbulence with mean-flow three-dimensionality is ubiquitous in geophysical and engineering flows. Under these conditions, turbulence may experience a counter-intuitive depletion of the turbulent stresses, which has important implications for modelling and control. Yet, current turbulence theories have been established mainly for statistically two-dimensional equilibrium flows and are unable to predict the reduction in the Reynolds stress magnitude. In the present work, we propose a multiscale model which explains the response of non-equilibrium wall-bounded turbulence under the imposition of three-dimensional strain. The analysis is performed via direct numerical simulation of transient three-dimensional turbulent channels subjected to a sudden lateral pressure gradient at friction Reynolds numbers up to 1,000. We show that the flow regimes and scaling properties of the Reynolds stress are consistent with a model comprising momentum-carrying eddies with sizes and time scales proportional to their distance to the wall. We further demonstrate that the reduction in Reynolds stress follows a spatially and temporally self-similar evolution caused by the relative horizontal displacement between the core of the momentum-carrying eddies and the flow layer underneath. Inspection of the flow energetics reveals that this mechanism is associated with lower levels of pressurestrain correlation which ultimately inhibits the generation of Reynolds stress. Finally, we assess the ability of the state-of-the-art wall-modelled large-eddy simulation to predict non-equilibrium, three-dimensional flows.