Theory Of Coherent Structures Research Articles

Lagrangian coherent structures and material transport in unsteady flow of vertical-axis turbine wakes

In this study, the unsteady phenomena of vertical-axis turbine wake flow were analyzed in two rotational speed conditions from a Lagrangian perspective. The wake flow field was obtained using particle image velocimetry. The spectral proper orthogonal decomposition method was used to decompose the wake modes with different frequencies. The Lagrangian coherent structure theory based on the finite-time Lyapunov exponent was used to study the material transport and mixing in the wake evolution process. The results show that the Lagrangian coherent structures can capture the boundary of the wake vortex. At a low rotation speed, a vorticity–flux window is found, through which the fluid is transported to the rotation-induced structures. At a high rotation speed, the increasing material transport between the separation bubbles and the incoming flow eventually leads to shedding of the large-scale vortex structures, accompanied by a decrease in the length of the separation zone. The material transport and mixing revealed in the unsteady flow of turbine wakes have significance in guiding the study of the flow control mechanism.

Local causal states and discrete coherent structures.

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the local causal states, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions.

Free-stream coherent structures in a planar jet

The free-stream coherent structure theory developed by Deguchi & Hall (J. Fluid Mech., vol. 752, 2014, pp. 602–625), valid in the large-Reynolds-number asymptotic limit, is extended and applied to jet flows. It is shown that a nonlinear exact coherent structure can be supported at the edge of the jet, and the structure induces a much bigger streaky flow in the centre of the jet. The lambda-shaped vortices that characterise the coherent structure are qualitatively consistent with those seen in experimental observations. Here a planar incompressible jet is investigated for the sake of simplicity, but the structure we describe could be used as a basis of more complex theories for incompressible and compressible jets of practical importance.

Binary interactions of solitary pulses in falling liquid films

We examine binary interaction of two-dimensional solitary pulses in falling liquid films. We make use of a two-field system of equations for the local flow rate and interface position that includes (second-order) viscous dispersion effects. By applying a coherent-structure theory, we obtain a dynamical system for the separation length between the pulses. The system is re-formulated in terms of a potential function which not only allows us to predict bound-state formation but also to interpret physically this formation. Numerical results of the fully non-linear second-order model are in very good agreement with the theoretically predicted distances. In addition, we numerically show that there are different types of dynamics associated with each bound state, including an overdamped, underdamped and undamped oscillating state depending on both the initial separation length and the Reynolds number.

Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization

We examine the interaction of two-dimensional solitary pulses on falling liquid films. We make use of the second-order model derived by Ruyer-Quil and Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000); Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a weighted residual technique. The model includes (second-order) viscous dispersion effects which originate from the streamwise momentum equation and tangential stress balance. These effects play a dispersive role that primarily influences the shape of the capillary ripples in front of the solitary pulses. We show that different physical parameters, such as surface tension and viscosity, play a crucial role in the interaction between solitary pulses giving rise eventually to the formation of bound states consisting of two or more pulses separated by well-defined distances and traveling at the same velocity. By developing a rigorous coherent-structure theory, we are able to theoretically predict the pulse-separation distances for which bound states are formed. Viscous dispersion affects the distances at which bound states are observed. We show that the theory is in very good agreement with computations of the second-order model. We also demonstrate that the presence of bound states allows the film free surface to reach a self-organized state that can be statistically described in terms of a gas of solitary waves separated by a typical mean distance and characterized by a typical density.

Many‐valued Logic in Multistate and Vague Stochastic Systems

Summary The state of the art in coherent structure theory is driven by two assertions, both of which are limiting: (1) all units of a system can exist in one of two states, failed or functioning; and (2) at any point in time, each unit can exist in only one of the above states. In actuality, units can exist in more than two states, and it is possible that a unit can simultaneously exist in more than one state. This latter feature is a consequence of the view that it may not be possible to precisely define the subsets of a set of states; such subsets are called vague. The first limitation has been addressed via work labeled ‘multistate systems’; however, this work has not capitalized on the mathematics of many‐valued propositions in logic. Here, we invoke its truth tables to define the structure function of multistate systems and then harness our results in the context of vagueness. A key contribution of this paper is to argue that many‐valued logic is a common platform for studying both multistate and vague systems but, to do so, it is necessary to lean on several principles of statistical inference.

Hysteresis phenomenon in turbulent convection

Coherent large-scale circulations of turbulent thermal convection in air have been studied experimentally in a rectangular box heated from below and cooled from above using Particle Image Velocimetry. The hysteresis phenomenon in turbulent convection was found by varying the temperature difference between the bottom and the top walls of the chamber (the Rayleigh number was changed within the range of $10^7 - 10^8$). The hysteresis loop comprises the one-cell and two-cells flow patterns while the aspect ratio is kept constant ($A=2 - 2.23$). We found that the change of the sign of the degree of the anisotropy of turbulence was accompanied by the change of the flow pattern. The developed theory of coherent structures in turbulent convection (Elperin et al. 2002; 2005) is in agreement with the experimental observations. The observed coherent structures are superimposed on a small-scale turbulent convection. The redistribution of the turbulent heat flux plays a crucial role in the formation of coherent large-scale circulations in turbulent convection.

Two-dimensional wave dynamics in thin films. II. Formation of lattices of interacting stationary solitary pulses

We develop an inelastic coherent structure theory that describes the weak interaction of the two-dimensional (2D) solitary pulses constructed in Part I. We focus on the interaction between two equilibrium pulses. We project the interaction dynamics on to the zero modes associated with the translational invariances of the system in the streamwise and spanwise directions and we obtain a dynamical system for the locations of the two solitons. We show that in the strongly dispersive case, 2D pulses self-organize into V shapes. Our theoretical findings are in excellent agreement with our time-dependent computations of the fully nonlinear system.

Statistical equilibrium predictions of jets and spots on Jupiter.

An equilibrium statistical theory of coherent structures is applied to midlatitude bands in the northern and southern hemispheres of Jupiter. The theory imposes energy and circulation constraints on the large-scale motion and uses a prior distribution on potential vorticity fluctuations to parameterize the small-scale turbulent eddies. Nonlinearly stable coherent structures are computed by solving the constrained maximum entropy principle governing the equilibrium states of the statistical theory. The theoretical predictions are consistent with the observed large-scale features of the weather layer if and only if the prior distribution has anticyclonic skewness, meaning that intense anticyclones predominate at small scales. Then the computations show that anticyclonic vortices emerge at the latitudes of the Great Red Spot and the White Ovals in the southern band, whereas in the northern band no vortices form within the zonal jets. Recent observational data from the Galileo mission support the occurrence of intense small-scale anticyclonic forcing. The results suggest the possibility of using equilibrium statistical theory for inverse modeling of the small-scale characteristics of the Jovian atmosphere from observed features.

Front interaction on a ring electrode

Recent electrochemical experiments and simulations by Krischer and co-workers on a ring electrode reveal a unique transition mechanism for a bistable reduction reaction. The front boundaries of a localized pulse of higher potential are seen to accelerate around the ring to induce a very rapid transition. We show here that the accelerated transition is due to attractive front interaction across the pulse initially and around the ring in the final stage. Using coherent structure theory, we quantitatively correlate this interaction-induced acceleration to the dimensions of the electrodes. We also predict a critical width of the initial pulse below which the pulse will shrink and the transition is prevented.

Local stability theory of solitary pulses in an active medium

If the spatio-temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses, its dynamics can often be described by a low-dimensional dynamical system. This welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction. It is shown that the structures can be destroyed by two types of local disturbance—an expanding radiation packet and a localized disturbance which travels with the structure. These two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized Kuramoto-Sivashinsky (gKS) equation in an extended domain. The possibility of describing pulse-radiation interaction by “resonance poles” also suggests that a more general coherent structure theory can be developed.

Interaction dynamics of solitary waves on a falling film

Beyond a short transition region near the inlet, waves on a falling film evolve into distinct pulse-like solitary waves that dominate all subsequent interfacial dynamics. Numerical and physical experiments indicate that these localized structures can attract and repel each other. Attractive interaction through the capillary ripples of the pulses causes two pulses to coalesce into a bigger pulse which accelerates and precipitates further coalescence. This binary interaction between an ‘excited’ pulse after coalescence and its smaller front neighbour is the key mechanism that drives the observed wave dynamics. From symmetry arguments, two dominant modes for a solitary pulse are obtained and used to develop an inelastic coherent structure theory for binary interaction between an excited pulse and its front neighbour. The theory offers a simple dynamical system that quantitatively describes the binary interaction and promises to elucidate the complex wave dynamics on a falling film.

Maximum entropy states for rotating vortex patches

The statistical equilibrium theory of coherent structures in two-dimensional turbulence is used to study the coalescence and symmetrization of rotating vortex patches. In this theory, most probable states, which are local probability distributions on the fluctuating vorticity field, are characterized by maximizing entropy subject to constraints derived from the conserved quantities for the Euler equations. The merger of a symmetric pair of patches and the axisymmetrization of an elliptical patch are computed by solving the constrained maximum entropy problem. In this way, these generic phenomena, which are usually simulated by direct time integration, are shown to be equilibration processes for the statistical mechanics model. Moreover, macroscopic features of the filamentation generated in these processes are captured by the statistical equilibrium theory. Namely, the delicate balance between the coalescence into a vortex core and the migration of filamentary vorticity away from the core, which is dictated by the conservation of angular impulse, is exhibited by the equilibrium solutions.

Effects of turbulent coherent structure on plume diffusion in atmosphere boundary layer

The simulation tests of plume diffusion have been carried out in an open water channel, in which a Laser Doppler Velocitymeter, optical concentration measurement and flow visualization technology have been used. Through experiments and theoretical analyses, the effects of environmental wind field, surface roughness and discharge condition of elevated source on plume diffusion have been investigated. The plume diffusion mechanism has been elucidated reasonably, based on the turbulent coherent structure theory.