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]

Abstract

S3T (Stochastic Structural Stability Theory) employs a closure at second order to obtain the dynamics of the statistical mean turbulent state. When S3T is implemented as a coupled set of equations for the streamwise mean and perturbation states, nonlinearity in the dynamics is restricted to interaction between the mean and perturbations. The S3T statistical mean state dynamics can be approximately implemented by similarly restricting the dynamics used in a direct numerical simulation (DNS) of the full Navier-Stokes equations (referred to as the NS system). Although this restricted nonlinear system (referred to as the RNL system) is greatly simplified in its dynamics in comparison to the associated NS, it nevertheless self-sustains a turbulent state in wall-bounded shear flow with structures and dynamics comparable to that in observed turbulence. Moreover, RNL turbulence can be analyzed effectively using theoretical methods developed to study the closely related S3T system. In order to better understand RNL turbulence and its relation to NS turbulence, an extensive comparison is made of diagnostics of structure and dynamics in these systems. Although quantitative differences are found, the results show that turbulence in the RNL system closely parallels that in NS and suggest that the S3T/RNL system provides a promising reduced complexity model for studying turbulence in wall-bounded shear flows. arXiv:1401.7816v1 [physics.flu-dyn]

Although the roll/streak structure is ubiquitous in both observations and simulations of pretransitional wall-bounded shear flow, this structure is linearly stable if the idealization of laminar flow is made. Lacking an instability, the large transient growth of the roll/streak structure has been invoked to explain its appearance as resulting from chance occurrence in the background turbulence of perturbations configured to optimally excite it. However, there is an alternative interpretation for the role of free-stream turbulence in the genesis of the roll/streak structure which is that the background turbulence interacts with the roll/streak structure to destabilize it. Statistical state dynamics (SSD) provides analysis methods for studying instabilities of this type which arise from interaction between the coherent and incoherent components of turbulence. Stochastic structural stability theory (S3T), which implements SSD in the form of a closure at second order, is used in this work to analyze the SSD modes arising from interaction between the coherent streamwise invariant component and the incoherent background component of turbulence. In pre-transitional Couette flow a manifold of stable modes with roll/streak form is found to exist in the presence of low intensity background turbulence. The least stable mode of this manifold is destabilized at a critical value of a parameter controlling the background turbulence intensity and a finite amplitude roll/streak structure arises from this instability through a bifurcation in this parameter. Although this bifurcation has analytical expression only in SSD, it is closely reflected in both the dynamically similar quasi-linear system, referred to as the restricted non-linear (RNL) system, and in DNS. This correspondence is verified using ensemble implementations of the RNL and DNS systems. S3T also predicts a second bifurcation at a higher value of the turbulent excitation parameter that results in destabilization of the finite amplitude roll/streak equilibria. This second bifurcation is shown to lead first to time dependence of the roll/streak in the S3T system and then to chaotic fluctuation corresponding to minimal channel turbulence. This transition scenario is also verified in simulations of the RNL and DNS systems. Bifurcation from a finite amplitude roll/streak equilibrium provides a direct route to the turbulent state through the S3T roll/streak instability. arXiv:1607.05018v3 [physics.flu-dyn]

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