Session A11: Nonlinear Dynamics: Coherent Structures

Explaining flow patterns by non-existing solutions of the governing equations

Invariant solutions of the governing equations, such as unstable equilibria and periodic orbits, are believed to serve as elementary building blocks of chaotic fluid flows and to play a major role in the emergence of patterns and coherent flow structures. Close to a saddle-node bifurcation, when two invariant solutions collide and annihilate, the flow behavior can closely resemble that of the solution at the bifurcation point, even though the solution itself does not exist at the studied parameter value. Therefore, patterns and coherent flow structures may emerge as a result of the dynamics feeling a non-existing invariant solution, a phenomenon called the ‘ghost’ of a solution. 

We use recently developed variational methods to formalize the concept of ghosts, follow ghost states in parameter space and demonstrate how even non-existing invariant solutions may control the behaviour of the chaotic flows, including 3D Rayleigh-B\'enard convection.   

Using exact coherent structures to describe intermittent Taylor-Couette flow

A dynamical systems approach to understanding turbulence suggests that the complicated motion of turbulent flow is shaped by special solutions of the Navier-Stokes equations known as exact coherent structures (ECSs). In this picture, turbulent flow co-evolves with, or "shadows", at least one such solution for a time; then a different ECS; and so on. Here we describe how ECSs can characterize flow in a Taylor-Couette experiment exhibiting temporal intermittency with periods of high regularity (quiescence) alternating with time intervals of significant spatial and temporal irregularity (activity). A collection of ECSs is found that describes this behavior for almost the entire duration of the flow, in both experimental and numerical investigations, for both quiescent and active periods of the flow. It is also demonstrated that the transition between quiescence and activity is mediated by a specific ECS that connects both regions. Finally, observables of the flow are shown to be well-represented by weighted averages of ECS observables, thereby demonstrating the connection between dynamical shadowing of ECSs and average observables in a 3-dimensional experimental flow.   

Transition to turbulence in the Stokes Boundary Layer: Edge States and Unsteady Self-Sustained Process (USSP)

The Stokes boundary layer is an oscillatory flow above an infinite plate, with oscillations driven either by (1) a transverse sinusoidal motion of the plate or (2) a sinusoidal applied pressure gradient. Beyond a critical Reynolds number of 2511, the laminar solution of the Stokes boundary layer is susceptible to linear instability. However, this instability is subcritical given that turbulence is observed for Reynolds numbers above approximately 700 despite the flow being linearly stable in this range. 

The state space of a subcritical flow consists of two basins of attraction: that of the laminar flow and that of turbulence. A saddle point separates these basins, termed the ‘edge state’, and its codimension-one stable manifold termed the ‘edge manifold’, or simply ‘edge’. The edge states may be found by ‘edge tracking’, an iterative procedure in which the trajectories of initial conditions on either side of the edge are computed and bisected.

Edge states in the Stokes boundary layer are composed of vortical structures of the same nature as canonical shear flows, namely streaks, rolls and waves. For non-oscillating shear flows, these structures are known to coexist and mutually balance through a mechanism known as the Self-Sustained Process. However, in the Stokes boundary layer, these structures are inherently periodic and utilise the oscillating base flow in a novel way. Structures migrate upwards to align with the location of the maximum shear of the laminar velocity profile, and large-scale rolls form to periodically create new streaks near the wall. This talk will present these edge states in the Stokes boundary layer, compare them to their known non-oscillatory counterparts, and provide insights about their effects on mass and momentum transport near the wall.   

Coherent structure interactions driven by excited hidden modes

Understanding the dynamics and interaction of coherent structures, such as solitary pulses, is an active topic of fundamental research in nonlinear science. It is well known that a group of interacting pulses may lead to a variety of dynamical regimes, from well-organized steady states to spatio-temporal chaos. An example is a liquid film flowing down a vertical/inclined substrate, a relatively simple open-flow hydrodynamic system, ubiquitous in many engineering applications, such as heat exchangers and chemical reactor columns. This system exhibits a rich variety of spatio-temporal structures that are generic to a large class of hydrodynamic and other nonlinear systems. 

Several studies have looked into the dynamics and interactions of pulses. The basic idea is to assume a superposition of interacting pulses, allowing for a "multiparticle description" of the system. However, previous analyses, including our own, focused on weakly interacting pulses, and little is known about dynamic states characterized by strongly interacting pulses, a challenging problem. The purpose of this work is precisely to analyze and quantify strong interactions between coherent structures in falling films. In this direction, we put forward a new coherent structure interaction framework encompassing both strong and weak interactions, underpinning also the previous theories on weak interaction. Specifically, we show that under some conditions a system of two pulses undergoes a transition from a regime of decaying oscillatory dynamics to self-sustained oscillations. Detailed examination of this transition shows that it is governed by a novel mechanism---a peculiar and unusual Hopf bifurcation in which a hidden complex conjugate resonance pair crosses the imaginary axis in the complex plane. We show that such a resonance pair results from the splitting of the resonance pole of the single-pulse system. This is a hidden object that is not part of the spectrum of the linearised operator, and only becomes visible when using appropriate weighted functional spaces. Spectral properties similar to those of the falling film problem, responsible for the existence of resonance poles, are prevalent in a wide class of active-dissipative systems, ranging from fluid dynamics to reaction-diffusion systems.   

Evaluating the complexity of a separated laminar boundary layer flow using the flow's topological features and excess entropy

Unsteady flows are notoriously complex, yet the definition of flow complexity remains vague. A complex flow may contain numerous interacting structures, with highly unpredictable dynamic behavior. Some studies describe flow complexity from a multi-scale modeling perspective. However, limited effort has been made to evaluate a flow’s degree-of-complexity (DOC). This presentation addresses the issue using a new information theory-based framework. A separated laminar boundary layer flow is examined with time-resolved PIV. The flow evolution is divided into multiple pseudo-periods, with the velocity in each period converted into a phase trajectory. These trajectories are then categorized and symbolized based on their homological group. The resulting symbol sequence allows for the evaluation of DOC using the well-established excess entropy method. Along the streamwise direction, the flow’s complexity shows a sudden transition near the separation point. Downstream of the separation, the DOC resembles that of a random signal, indicating that the flow structures have no correlation over time.   

Liquid-liquid phase-separated pattern propagation in ternary mixtures

Ternary mixtures can undergo liquid-liquid phase separation in response to concentration changes. It is observed that a ternary mixture of oil-water-ethanol in a microchannel with water and surfactant leads to the formation of a phase separating front, leaving alternating oil- and water-rich stripes in its wake due to ethanol diffusion out of the mixture [Moerman et al. PNAS 2018, 115 (12), 3599-3604]. We model these dynamics via a system with an initially stable ternary mixture (oil-water-ethanol) and a stable single-component phase (water) in contact. Letting ethanol preferentially diffuse out of the ternary mixture causes the mixture to undergo spinodal decomposition from the interface. We recast the Flory-Huggins free energy – assuming that the interaction parameters involving ethanol are zero – to get an effective binary mixture description, parameterized by ethanol concentration. This makes the ternary mixture likely to become unstable below a cut-off ethanol concentration. Using Cahn-Hilliard dynamics, we explore features of diffusion-mediated phase-separated patterns such as length scale, phase composition, and front velocity. Extension of this idea can help understand how to control phase separation in co-flow systems, where advection can affect phase-separated patterns.   

Non-linear hydrodynamic forces in opposite micro cantilever beams oscillating in viscous fluid

The present study investigates the nonlinear hydrodynamic forces in opposing microcantilever beams oscillating in a viscous fluid. Using FEM-based models solves complete Navier Stokes equations, we estimate the nonlinear damping and added mass forces at high oscillation frequencies due to convetive inertial forces and vortex formation at edges of the beams, considering both in-phase and out-of-phase conditions. Additionally, we calculate the inertial and drag forces using the semi-analytical boundary element method (BEM) by incorporating the unsteady Stokes equation. Our findings indicate a strong correlation between numerical and semi-analytical BEM results for small amplitude and up to 100kHz of frequency. This research is particularly significant for MEMS devices, where maintaining small gaps between opposing or substrate beams is crucial. We focus on key aspects such as added mass effects and damping effects (Quality factor), using both methods to compute the hydrodynamic coupling effects while ensuring no influence from surrounding walls. Our study reveals that the maximum effective gap for significant hydrodynamic coupling effects is 15 micrometers; beyond this gap, the coupling effects become negligible and sensitivity of the structre is invarient. Furthermore, we extend our analysis to non-uniform beams to explore nonlinear damping at small oscillation amplitudes, providing a comprehensive understanding of the dynamic behaviors in these systems.