Session LB: Chaos and Fractals I

Driven Lid Flow in a Hemisphere: Steady-State Stokes Flow and 3-Dimensional Chaotic Advection Under Small and Large Perturbation

We investigate flow and transport in a hemisphere filled with viscous fluid whose equitorial plane is a sliding lid. For steady-state Stokes flow the hemisphere has a single vortex line, nearly wall-attached, perpendicular to the sliding direction. Passive particle transport in the hemisphere is confined to cigar shaped shells that are continuously deformed from the vortex line to the boundary. We perturb the steady flow in two ways. (1) A small amplitude azimuthal oscillation in the sliding direction of the lid creates a fully three-dimensional dynamical system for particle transport in the hemisphere. Orbits between shells are connected through close approach manifolds at the parabolic points on the wall. (2) A periodically reoriented flow (PRF) occurs when the lid slides in one direction for a finite time $\tau$ then slides (without twisting) in a new direction reoriented from the first by a finite angle $\Theta$. The PRF introduces symmetry, and we search for a mixing optimum in this stirred PRF.   

Moving Walls Accelerate Mixing

Mixing in viscous fluids is challenging, but chaotic advection in principle allows efficient mixing. In the best possible scenario, the decay rate of the concentration profile of a passive scalar should be exponential in time. In practice, several authors have found that the no-slip boundary condition at the walls of a vessel can slow down mixing considerably, turning an exponential decay into a power law. This slowdown affects the whole mixing region, and not just the vicinity of the wall. The reason is that when the ergodic mixing region extends to the wall, a separatrix connects to it. The approach to the wall along that separatrix is polynomial in time and dominates the long-time decay. However, if the walls are moving then closed orbits are created, separated from the bulk by a homoclinic orbit connected to a hyperbolic fixed point. The long-time approach to the fixed point is exponential, so we recover an overall exponential decay, albeit with a thin unmixed region near the wall.   

The effect of finite Reynolds numbers on chaotic advection

The effect of finite Reynolds numbers on chaotic advection is investigated for two dimensional lid-driven cavity flows that exhibit topological chaos in the creeping flow regime. The exponential convergence afforded by the use of spectral methods enables accurate tracking of passive scalars and exponential stretching of material lines in the flow. The emphasis in this endeavor is to study how the inertial effects present due to the low, but non-zero, Reynolds numbers produce irreversibilities in the system that affect the efficacy of mixing. Comparisons are made with Stokes flow results for the same configurations. The application of these results to systems that can effectively separate two substances with slightly different diffusivities in a medium are discussed.   

On the role of topological chaos and ghost rods in fluid mixing

We consider stirring and mixing of two-dimensional Stokes flow in a circular domain due to the motion of three rods. Two similar protocols are discussed that are expected to give significantly different results based on the predictions of the Thurston-Nielsen (TN) theorem. Somewhat surprisingly, under many conditions the topologically ``trivial'' finite order protocol produces a larger stretch rate than does the pseudo-Anosov protocol, which is guaranteed to be chaotic by the TN theorem. We show that, in these cases, periodic points in the flow act as ``ghost rods'' that can be considered responsible for the large stretch rates produced by the finite order protocol. However, the existence and importance of these ghost rods is dependent on the specific system geometry, and perturbations can lead to very low stretch rates when using the finite order protocol. In contrast, selection of a pseudo-Anosov protocol leads to a robust minimum for the stretch rates as predicted by the TN theorem. In order to associate the stretch rate results to fluid mixing, we also discuss the homogenization of a passive scalar advected by the flow.   

Stirring with ghost rods in a lid-driven cavity

It has shown that passive fluid particles moving on periodic orbits can be used to `stir' a viscous fluid in a two-dimensional lid-driven cavity that exhibits a figure-eight flow pattern (Stremler \& Chen 2007). Fluid motion in the vicinity of these particles produces ``ghost rod'' structures that behave like semi-permeable rods in the flow. Since these ghost rods are present due to the system dynamics, perturbations in the boundary conditions lead to variations in the existence and structure of the ghost rods. We discuss these variations and assess the role of ghost rods in mixing over a range of operating conditions for this system. The results suggest that ghost rods can play an important role in mixing for other counter-rotating flows.   

Almost-invariant sets as ``ghost rods'' for fluid stirring

In two-dimensional time-dependent flows or three-dimensional flows with a certain symmetry, the braiding of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen (TN) classification theorem. ``Ghost rods,'' or periodic orbits generated by the dynamics, behave as physical obstructions that ``stir'' the surrounding fluid, and these can be used as the basis for this topological analysis. We explore the identification of almost-invariant sets, or regions of fluid with high local residence time, as ghost rods. This set-oriented approach can be applied using relatively coarse system information, making this a promising approach for extending the use of the TN classification theorem to a variety of fluid systems.   

Fast computation of Lagrangian coherent structures: algorithms and error analysis

This work investigates a number of efficient methods for computing finite time Lyapunov exponent (FTLE) fields in unsteady flows by approximating the particle flow map and eliminating redundant particle integrations in neighboring flow maps. Ridges of the FTLE fields are Lagrangian coherent structures (LCS) and provide an unsteady analogue of invariant manifolds from dynamical systems theory. The fast methods fall into two categories, unidirectional and bidirectional, depending on whether flow maps in one or both time directions are composed to form an approximate flow map. An error analysis is presented which shows that the unidirectional methods are accurate while the bidirectional methods have significant error which is aligned with the opposite time coherent structures. This relies on the fact that material from the positive time LCS attracts onto the negative time LCS near time-dependent saddle points.   

Topology counts: Statistics of critical points in experimental, two-dimensional flow

Points in a flowing fluid where the speed is zero~--- and therefore no streamline can be drawn~--- are known as critical points and have special topological significance. Two types exist in two-dimensional flows: hyperbolic (saddle) points and elliptic (center) points. Approximating two-dimensional flow with an electromagnetically driven, stably stratified solution in a 90~cm x 90~cm tray, we use particle tracking to measure the velocity field and locate the critical points. Our field of view encompasses $\sim$200 critical points per frame, each of which can be tracked like a particle over many frames. We will discuss the resulting spatiotemporal statistics of critical points in two-dimensional flow, focusing in particular on number fluctuations.   

Chaotic mixing in vortex-dominated flows

We present experimental studies of chaotic mixing in time-periodic, two-dimensional (2D) arrays of vortices. The flows are formed from the superposition of two vortex arrays shifted by half a vortex width in both directions. Flows generated by this method show both diffusive and superdiffusive transport, depending on the relative strengths and nature of the time dependence of the currents producing the two vortex arrays. Experimentally, we track the motion of tracer particles moving with the flow. From these tracks, we can determine the growth of the variance of a distribution of tracers. We are also applying algorithms based on braiding analysis \footnote{J.-L. Thiffeault, preprint} to determine the topological entropy for mixing in these flows. We are also initiating studies of chaotic mixing in a 3D, time-independent flow (composed of nested vortices) with the goal of studying advection-reaction-diffusion processes in a 3D system.   

Burning lobes in an advection-reaction-diffusion system

We use the tools developed for chaotic fluid transport to describe the propagation of a reaction front in a chain of oscillating vortices. Specifically, we expand the concept of lobes (turnstiles) bounded by stable and unstable manifolds to account for the propagation of a reaction front. A ``burning lobe'' is a passive lobe that grows due to the reactive medium. We propose that a reaction front will propagate from one vortex to the next in an oscillation period if a portion of the front was within the burning lobe. This framework is used to explain mode-locking of reaction fronts propagating in an oscillating vortex array.