Distinct organizational States of fully developed turbulent pipe flow

Simultaneous Eulerian-Lagrangian velocity measurements of particulate pipe flow in transitional regime

We present a unique pipe flow rig capable of simultaneous particle tracking and flow velocity measurements in a dilute, neutrally buoyant particulate pipe flow in regimes of transition to turbulence. The flow consists of solid glass spheres for the disperse phase and a density-matching fluid for the carrier phase. The measurements are conducted using a bespoke, combined two-dimensional particle image velocimetry and particle tracking velocimetry technique. The technique takes advantage of a phase discrimination approach that involves separating the disperse and carrier phases based on their respective image characteristics. Our results show that the rig and the technique it implements can effectively be employed to study transitional particulate pipe flows at dilute concentrations.

Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.

Unstable equilibria and invariant manifolds in quasi-two-dimensional Kolmogorov-like flow

Recent studies suggest that unstable, nonchaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space.

Forecasting Fluid Flows Using the Geometry of Turbulence

The existence and dynamical role of particular unstable solutions (exact coherent structures) of the Navier-Stokes equation is revealed in laboratory studies of weak turbulence in a thin, electromagnetically driven fluid layer. We find that the dynamics exhibit clear signatures of numerous unstable equilibrium solutions, which are computed using a combination of flow measurements from the experiment and fully resolved numerical simulations. We demonstrate the dynamical importance of these solutions by showing that turbulent flows visit their state space neighborhoods repeatedly. Furthermore, we find that the unstable manifold associated with one such unstable equilibrium predicts the evolution of turbulent flow in both experiment and simulation for a considerable period of time.

Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence

We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just a few modes of the model of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)]. This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence.

Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

Coherent structures in wall-bounded turbulence

The inherent difficulty of understanding turbulence has led to researchers attacking the topic in many different ways over the years of turbulence research. Some approaches have been more successful than others, but most only deal with part of the problem. One approach that has seen reasonable success (or at least popularity) is that of attempting to deconstruct the complex and disorganised turbulent flow field into to a set of motions that are in some way organised. These motions are generally called "coherent structures". There are several strands to this approach, from identifying the coherent structures within the flow, defining their characteristics, explaining how they are created, sustained and destroyed, to utilising their features to model the turbulent flow. This review considers research on coherent structures in wall-bounded turbulent flows: a class of flow which is extremely interesting to many scientists (mainly, but not exclusively, physicists and engineers) due to their prevalence in nature, industry and everyday life. This area has seen a lot of activity, particularly in recent years, much of which has been driven by advances in experimental and computational techniques. However, several ideas, developed many years ago based on flow visualisation and intuition, are still both informative and relevant. Indeed, much of the more recent research is firmly indebted to some of the early pioneers of the coherent structures approach. Therefore, in this review, selected historical research is discussed along with the more contemporary advances in an attempt to provide the reader with a good overview of how the field has developed and to highlight the perspicacity of some of the early researchers, as well as providing an overview of our current understanding of the role of coherent structures in wall-bounded turbulent flows.