Reduced description of exact coherent states in parallel shear flows

Following marginal stability manifolds in quasilinear dynamical reductions of multiscale flows in two space dimensions

We derive a two-dimensional (2D) extension of a recently developed formalism for slow-fast quasilinear (QL) systems subject to fast instabilities. The emergent dynamics of these systems is characterized by a slow evolution of (suitably defined) mean fields coupled to marginally stable, fast fluctuation fields. By exploiting this scale separation, an efficient hybrid fast-eigenvalue/slow-initial-value solution algorithm can be developed in which the amplitude of the fast fluctuations is slaved to the slowly evolving mean fields to ensure marginal stability-and temporal scale separation-is maintained. For 2D systems, the fluctuation eigenfunctions are labeled by their Fourier wave numbers characterizing spatial variability in that extended spatial direction, and the marginal mode(s) must coincide with the fastest-growing mode(s) over all admissible Fourier wave numbers. Here we derive an ordinary differential equation governing the slow evolution of the wave number of the fastest-growing fluctuation mode that simultaneously must be slaved to the mean dynamics to ensure the mode has zero growth rate. We illustrate the procedure in the context of a 2D model partial differential equation that shares certain attributes with the equations governing strongly stratified shear flows and other strongly constrained forms of geophysical turbulence in extreme parameter regimes. The slaved evolution follows one or more marginal stability manifolds, which constitute select state-space structures that are not invariant under the full flow dynamics yet capture quasicoherent structures in physical space in a manner analogous to invariant solutions identified in, e.g., transitionally turbulent shear flows. Accordingly, we propose that marginal stability manifolds are central organizing structures in a dynamical systems description of certain classes of multiscale flows in which scale separation justifies a QL approximation of the dynamics.

Training a neural network to predict dynamics it has never seen

Neural networks have proven to be remarkably successful for a wide range of complicated tasks, from image recognition and object detection to speech recognition and machine translation. One of their successes lies in their ability to predict future dynamics given a suitable training data set. Previous studies have shown how echo state networks (ESNs), a type of recurrent neural networks, can successfully predict both short-term and long-term dynamics of even chaotic systems. This study shows that, remarkably, ESNs can successfully predict dynamical behavior that is qualitatively different from any behavior contained in their training set. Evidence is provided for a fluid dynamics problem where the flow can transition between laminar (ordered) and turbulent (seemingly disordered) regimes. Despite being trained on the turbulent regime only, ESNs are found to predict the existence of laminar behavior. Moreover, the statistics of turbulent-to-laminar and laminar-to-turbulent transitions are also predicted successfully. The utility of ESNs in acting as early-warning generators for transition is discussed. These results are expected to be widely applicable to data-driven modeling of temporal behavior in a range of physical, climate, biological, ecological, and finance models characterized by the presence of tipping points and sudden transitions between several competing states.

Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent-incoherent interaction is lumped into a Reynolds averaged Navier-Stokes (RANS) eddy viscosity model, and the mean-coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793, 777-797. (doi:10.1017/jfm.2016.109)). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

A self-sustaining process model of inertial layer dynamics in high Reynolds number turbulent wall flows

Field observations and laboratory experiments suggest that at high Reynolds numbers Re the outer region of turbulent boundary layers self-organizes into quasi-uniform momentum zones (UMZs) separated by internal shear layers termed 'vortical fissures' (VFs). Motivated by this emergent structure, a conceptual model is proposed with dynamical components that collectively have the potential to generate a self-sustaining interaction between a single VF and adjacent UMZs. A large-Re asymptotic analysis of the governing incompressible Navier-Stokes equation is performed to derive reduced equation sets for the streamwise-averaged and streamwise-fluctuating flow within the VF and UMZs. The simplified equations reveal the dominant physics within-and isolate possible coupling mechanisms among-these different regions of the flow.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

Prospectus: towards the development of high-fidelity models of wall turbulence at large Reynolds number

Recent and on-going advances in mathematical methods and analysis techniques, coupled with the experimental and computational capacity to capture detailed flow structure at increasingly large Reynolds numbers, afford an unprecedented opportunity to develop realistic models of high Reynolds number turbulent wall-flow dynamics. A distinctive attribute of this new generation of models is their grounding in the Navier-Stokes equations. By adhering to this challenging constraint, high-fidelity models ultimately can be developed that not only predict flow properties at high Reynolds numbers, but that possess a mathematical structure that faithfully captures the underlying flow physics. These first-principles models are needed, for example, to reliably manipulate flow behaviours at extreme Reynolds numbers. This theme issue of Philosophical Transactions of the Royal Society A provides a selection of contributions from the community of researchers who are working towards the development of such models. Broadly speaking, the research topics represented herein report on dynamical structure, mechanisms and transport; scale interactions and self-similarity; model reductions that restrict nonlinear interactions; and modern asymptotic theories. In this prospectus, the challenges associated with modelling turbulent wall-flows at large Reynolds numbers are briefly outlined, and the connections between the contributing papers are highlighted.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.