Turbulence tracks recurrent solutions.

Despite a long and rich history of scientific investigation, fluid turbulence remains one of the most challenging problems in science and engineering. One of the key outstanding questions concerns the role of coherent structures that describe frequently observed patterns embedded in turbulence. It has been suggested, but not proved, that coherent structures correspond to unstable, recurrent solutions of the governing equation of fluid dynamics. Here, we present experimental and numerical evidence that three-dimensional turbulent flow tracks, episodically but repeatedly, the spatial and temporal structure of multiple such solutions. Our results provide compelling evidence that coherent structures, grounded in the governing equations, can be harnessed to predict how turbulent flows evolve.

Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression.

Machine learning offers an intriguing alternative to first-principle analysis for discovering new physics from experimental data. However, to date, purely data-driven methods have only proven successful in uncovering physical laws describing simple, low-dimensional systems with low levels of noise. Here we demonstrate that combining a data-driven methodology with some general physical principles enables discovery of a quantitatively accurate model of a non-equilibrium spatially extended system from high-dimensional data that is both noisy and incomplete. We illustrate this using an experimental weakly turbulent fluid flow where only the velocity field is accessible. We also show that this hybrid approach allows reconstruction of the inaccessible variables - the pressure and forcing field driving the flow.

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.

Heteroclinic and homoclinic connections in a Kolmogorov-like flow.

Recent studies suggest that unstable recurrent solutions of the Navier-Stokes equation provide new insights into dynamics of turbulent flows. In this study, we compute an extensive network of dynamical connections between such solutions in a weakly turbulent quasi-two-dimensional Kolmogorov flow that lies in the inversion-symmetric subspace. In particular, we find numerous isolated heteroclinic connections between different types of solutions-equilibria, periodic, and quasiperiodic orbits-as well as continua of connections forming higher-dimensional connecting manifolds. We also compute a homoclinic connection of a periodic orbit and provide strong evidence that the associated homoclinic tangle forms the chaotic repeller that underpins transient turbulence in the symmetric subspace.

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.

Exploring physics students' engagement with online instructional videos in an introductory mechanics course.

The advent of new educational technologies has stimulated interest in using online videos to deliver content in university courses. We examined student engagement with 78 online videos that we created and were incorporated into a one-semester flipped introductory mechanics course at the Georgia Institute of Technology. We found that students were more engaged with videos that supported laboratory activities than with videos that presented lecture content. In particular, the percentage of students accessing laboratory videos was consistently greater than 80% throughout the semester. On the other hand, the percentage of students accessing lecture videos dropped to less than 40% by the end of the term. Moreover, the fraction of students accessing the entirety of a video decreases when videos become longer in length, and this trend is more prominent for the lecture videos than the laboratory videos. The results suggest that students may access videos based on perceived value: students appear to consider the laboratory videos as essential for successfully completing the laboratories while they appear to consider the lecture videos as something more akin to supplemental material. In this study, we also found that there was little correlation between student engagement with the videos and their incoming background. There was also little correlation found between student engagement with the videos and their performance in the course. An examination of the in-video content suggests that students engaged more with concrete information that is explicitly required for assignment completion (e.g., actions required to complete laboratory work, or formulas or mathematical expressions needed to solve particular problems) and less with content that is considered more conceptual in nature. It was also found that students' in-video accesses usually increased toward the embedded interaction points. However, students did not necessarily access the follow-up discussion of these interaction points. The results of the study suggest ways in which instructors may revise courses to better support student learning. For example, external intervention that helps students see the value of accessing videos may be required in order for this resource to be put to more effective use. In addition, students may benefit more from a clicker question that reiterates important concepts within the question itself, rather than a clicker question that leaves some important concepts to be addressed only in the discussion afterwards.

Modal spectra extracted from nonequilibrium fluid patterns in laboratory experiments on Rayleigh-Bénard convection.

We describe a method to extract from experimental data the important dynamical modes in spatiotemporal patterns in a system driven out of thermodynamic equilibrium. Using a novel optical technique for controlling fluid flow, we create an experimental ensemble of Rayleigh-Bénard convection patterns with nearby initial conditions close to the onset of secondary instability. An analysis of the ensemble evolution reveals the spatial structure of the dominant modes of the system as well as the corresponding growth rates. The extracted modes are related to localized versions of instabilities found in the ideal unbounded system. The approach may prove useful in describing instability in experimental systems as a step toward prediction and control.

Extensive scaling from computational homology and Karhunen-Loève decomposition analysis of Rayleigh-Bénard convection experiments.

Spatiotemporally chaotic dynamics in laboratory experiments on convection are characterized using a new dimension, D(CH), determined from computational homology. Over a large range of system sizes, D(CH) scales in the same manner as D(KLD), a dimension determined from experimental data using Karhuenen-Loéve decomposition. Moreover, finite-size effects (the presence of boundaries in the experiment) lead to deviations from scaling that are similar for both D(CH) and D(KLD). In the absence of symmetry, D(CH) can be determined more rapidly than D(KLD).

Transient turbulence in Taylor-Couette flow.

Recent studies have brought into question the view that at sufficiently high Reynolds number turbulence is an asymptotic state. We present direct observation of the decay of turbulent states in Taylor-Couette flow with lifetimes spanning five orders of magnitude. We also show that there is a regime where Taylor-Couette flow shares many of the decay characteristics observed in other shear flows, including Poisson statistics and the coexistence of laminar and turbulent patches. Our data suggest that for a range of Reynolds numbers characteristic decay times increase superexponentially with increasing Reynolds number but remain bounded in agreement with the most recent data from pipe flow. Our data are also consistent with recent theoretical predictions of lifetime scaling in transitional flows.

State and parameter estimation of spatiotemporally chaotic systems illustrated by an application to Rayleigh-Bénard convection.

Data assimilation refers to the process of estimating a system's state from a time series of measurements (which may be noisy or incomplete) in conjunction with a model for the system's time evolution. Here we demonstrate the applicability of a recently developed data assimilation method, the local ensemble transform Kalman filter, to nonlinear, high-dimensional, spatiotemporally chaotic flows in Rayleigh-Bénard convection experiments. Using this technique we are able to extract the full temperature and velocity fields from a time series of shadowgraph measurements. In addition, we describe extensions of the algorithm for estimating model parameters. Our results suggest the potential usefulness of our data assimilation technique to a broad class of experimental situations exhibiting spatiotemporal chaos.

Chaotic mixing in microdroplets.

We describe a general methodology for introducing thorough chaotic mixing in microdroplets. The mixing properties of fluid flows in microdroplets are governed by their symmetries, which give rise to invariant surfaces serving as barriers to transport. Complete three-dimensional mixing by chaotic advection requires destruction of all flow invariants. To illustrate this idea, we demonstrate that complete mixing can be obtained in a time-dependent flow produced by moving a microdroplet along a two-dimensional path. The theoretical predictions are confirmed by experiments that use the thermocapillary effect to manipulate microdroplets.

Complex-ordered patterns in shaken convection.

We report and analyze complex patterns observed in a combination of two standard pattern forming experiments. These exotic states are composed of two distinct spatial scales, each displaying a different temporal dependence. The system is a fluid layer experiencing forcing from both a vertical temperature difference and vertical time-periodic oscillations. Depending on the parameters these forcing mechanisms produce fluid motion with either a harmonic or a subharmonic temporal response. Over a parameter range where these mechanisms have comparable influence the spatial scales associated with both responses are found to coexist, resulting in complex, yet highly ordered patterns. Phase diagrams of this region are reported and criteria to define the patterns as quasiperiodic crystals or superlattices are presented. These complex patterns are found to satisfy four-mode (resonant tetrad) conditions. The qualitative difference between the present formation mechanism and the resonant triads ubiquitously used to explain complex-ordered patterns in other nonequilibrium systems is discussed. The only exception to quantitative agreement between our analysis based on Boussinesq equations and laboratory investigations is found to be the result of breaking spatial symmetry in a small parameter region near onset.

Secondary instabilities of hexagonal patterns in a Bénard-Marangoni convection experiment.

We have identified experimentally secondary instability mechanisms that restrict the stable band of wave numbers for ideal hexagons in Bénard-Marangoni convection. We use "thermal laser writing" to impose long wave perturbations of ideal hexagonal patterns as initial conditions and measure the growth rates of the perturbations. For epsilon=0.46 our results suggest a longitudinal phase instability limits stable hexagons at a high wave number while a transverse phase instability limits low wave number hexagons.

Optical manipulation of microscale fluid flow.

A novel optical method is used both to probe and to control dynamics in experiments on the spreading of microscale liquid films over solid substrates. The flow is manipulated by thermally induced surface-tension gradients that are regulated by controlling the absorption of light in the substrate. This approach permits, for the first time, the measurement of the dispersion relation for the well-known contact line instability; the measurements are compared with theoretical predictions from the slip model for spreading films. The experiments also demonstrate the use of feedback control to suppress instability. These results show that optical control can provide dynamically reconfigurable manipulations of fluid flow, thereby suggesting a general approach for constructing reprogrammable microfluidic devices.

Evolution of hexagonal patterns from controlled initial conditions in a Bénard-Marangoni convection experiment.

We report quantitative measurements of both wave number selection and defect motion in nonequilibrium hexagonal patterns. A novel optical technique ("thermal laser writing") is used to imprint initial patterns with selected characteristics in a Bénard-Marangoni convection experiment. Initial patterns of ideal hexagons are imposed to determine the band of stable pattern wave numbers while initial patterns containing an isolated penta-hepta defect are imprinted to study defect propagation directions and velocities. The experimental results are compared to recent theoretical predictions.