Session L3: Dynamical Systems and Chaos II

Burning invariant manifolds for propagating fronts in a chain of vortices

We present experimental studies of the behavior of reaction fronts in a chain of alternating vortices. The flow is produced by a magnetohydrodynamic forcing technique, and the fronts are produced by the ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. We introduce {\em burning invariant manifolds} (BIMs) which act as barriers to front propagation, similar to the role played by invariant manifolds as barriers to passive transport in two-dimensional flows. Unlike manifolds for passive transport, though, BIMs are one-sided barriers, passing either left- or right-going fronts but blocking the other. We show how the BIMs can be measured experimentally for both time-independent and time-periodic flows. The experimental results are compared to simulations based on a simplified numerical model of the flow.   

Lobe dynamics and front propagation in advection-reaction-diffusion systems

We consider the addition of reaction-diffusion dynamics to systems undergoing chaotic advection. This can be viewed as a simplified model of diverse systems such as combustion dynamics in a chaotic flow, microfluidic chemical reactors, and blooms of phytoplankton and algae. Recently, we have proposed that front propagation in these systems is strongly influenced by burning invariant manifolds (BIMs)---geometric structures analogous to traditional invariant manifolds for passive transport. Additionally, BIMs may be used to define tangle-like structures that support a version of lobe dynamics for front propagation. In this talk, we discuss the theory and structure of BIMs and demonstrate the modified lobe-dynamics. We also present a potential application of the lobe dynamics to the control of reactive flows.   

The geometry of mode-locked fronts in periodically driven advection-reaction-diffusion systems

We consider reaction-diffusion dynamics within a periodically driven fluid forming a linear chain of alternating vortices. Prior theoretical and experimental work on this system has demonstrated rich structure in the dynamics of the reaction front; for certain parameter values, the front will be mode locked to the external driving. Using dynamical systems theory, we relate the mode-locking behavior to the existence of relative periodic orbits (RPOs), and we show that the mode-locked front itself follows the profile of a ``burning invariant manifold'' (BIM)---a generalization of traditional invariant manifolds for passive transport, which incorporates the dynamics of front propagation. Together, the RPOs and BIMs provide clear criteria for the emergence and destruction of mode locking as well as an explanation of the front geometry.   

Barriers to reaction front propagation in a spatially random, time-independent flow

We present experimental studies of barriers, called {\em burning invariant manifolds} (BIMs), to front propagation in a spatially random, time-independent flow. We generate the flow with a magnetohydrodynamic technique that uses a DC current and a disordered pattern of permanent magnets. The velocity field is determined from this flow using particle tracking velocimetry, and reaction fronts are produced using the Ferroin-catalyzed Belousov-Zhabotinsky (BZ) chemical reaction. We use the experimental velocity field and a three-dimensional set of ODEs to predict from theory the location and orientation of BIMs. These predicted BIMs are found to match up well with the propagation barriers observed experimentally in the same flow using the BZ reaction. We explore the nature of BIMs as one-sided barriers, in contrast to invariant manifolds that act as barriers for passive transport in all directions. We also explore the role of projection singularities in the theory and how these singularities affect front behavior.   

Computation of Lagrangian Coherent Structures from their Variational Theory

We describe a computational algorithm for detecting hyperbolic Lagrangian Coherent Structures (LCS) from a recently developed variational theory [1]. In contrast to earlier approaches to LCS, our algorithm is based on exact mathematical theorems that render LCS as smooth parametrized curves, i.e., trajectories of an associated ordinary differential equation. The algorithm also filters out LCS candidates that are pure artifacts of high shear. We demonstrate the algorithm on two-dimensional flow models and on an experimentally measured turbulent velocity field.\\[4pt] [1] G. Haller, A variational theory of hyperbolic Lagrangian Coherent Structures,\emph{\ Physica D} \textbf{240} (2011) 574-598   

Geodesic Theory of Transport Barriers in Unsteady Flows

We introduce a unified approach to detecting finite-time Lagrangian transport barriers in two-dimensional unsteady flows with general time dependence. Seeking transport barriers as least deforming material lines, we obtain a variational formulation for such barriers. This variational problem turns out to be well-posed only for three types of transport barriers: hyperbolic barriers (generalized stable and unstable manifolds), elliptic barriers (generalized KAM curves or eddy boundaries), and parabolic barriers (generalized shear jets). Such barriers then coincide with minimal geodesics under an appropriate metric induced by the Cauchy-Green strain tensor on the initial configuration of the flow. The geodesics are obtained as a solution of an ordinary differential equation, and hence are available in a smooth, parametrized form. We show how these new results reveal previously unknown transport barriers in complex model flows and geophysical data sets.   

Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinsky equation

The Kuramoto-Sivashinsky (KS) equation is a paradigmatic model for a wide spectrum of systems exhibiting spatio-temporal chaos, such as a thin-liquid film falling down a vertical substrate. Here we deal with the noisy KS eqiation which can be derived for the falling film problem with a topographically random substrate. We examine the effects of additive noise in a regime close to the instability onset. We show that when the noise is highly degenerate, in the sense that it acts only on the first stable mode, the solution of the KS equation undergoes several transitions between different states, including a critical on-off intermittent state that is eventually stabilized as the noise strength is increased. Similar results are obtained with the Burgers equation, which has often been used as a prototype of one-dimensional turbulence. Such noise-induced transitions can be completely characterized through critical exponents, obtaining that both equations belong to the same universality class. The results of our numerical investigations are explained rigorously using multiscale techniques.   

A new mode reduction strategy applied to the generalized Kuramoto-Sivashinsky equation

The generalized Kuramoto-Sivashinsky (gKS) equation is one of the simplest prototypes modeling nonlinear active media with energy supply, energy dissipation and dispersion. Not surprisingly, it has been reported for a wide variety of physical settings, from reaction-diffusion systems, e.g. propagation of concentration waves and flame-front instabilities, to fluid dynamics, e.g. a viscous film flowing down an inclined plane. We undertake a combined theoretical-numerical study of the gKS equation. We first approximate it with a renormalization group equation. This approximation forms the basis for a non-standard stochastic mode reduction that guarantees optimality in the sense of maximal information entropy. Herewith, noise is rigorously defined in the reduced gKS equation and hence provides an analytical explanation for its origin. These derivations allow us to develop reliable numerical approximations to the gKS equation and a rigorous methodology on how to add noise. Interestingly, noise becomes increasingly important by decreasing the degrees of freedom in the discretization strategy.   

Torus Streamlines in the 3D Steady Lid-driven Cavity Flows

Streamlines of the incompressible vortical flows in three-dimensional rectangular cavities with different aspect ratios are numerically studied for several Reynolds numbers by using a combined compact finite difference (CCD) scheme with high accuracy and high resolution. The flow is driven by a lid moving tangentially with constant speed. Non-dimensional geometrical parameters of the cavity are the depth-to-width aspect ratio $\Gamma $ and the span-to-width aspect ratio $\Lambda $. The flow parameter is the Reynolds number Re. We study the flow structures in the square cavity ($\Gamma $=1) with the spanwise aspect ratios $\Lambda $=1 and 6.55 for Re from 100 to 400. Torus streamlines are obtained from the velocity field of the steady incompressible flow. Several other streamlines show chaotic behavior. They are equivalent to a non-autonomous Hamiltonian system of one-degree-of-freedom. In order to examine the features of the flow pattern with different parameters, we analyze the Poincare sections in the cross sections of cavities.   

Topological chaos in a lid driven cavity flow at finite Reynolds number

Topological chaos, or chaos that is guaranteed to exist in a system due to sufficiently complex motion of a few periodic orbits, has been demonstrated for creeping flow in a lid driven cavity. Nearly-periodic systems can by analyzed in a similar way based on the presence of Almost Invariant Sets (AIS) with similarly complex space-time trajectories. We extend this analysis to finite Reynolds number flows in a lid driven cavity using a 2D Fourier-Chebyshev spectral algorithm for the streamfunction-vorticity formulation, which enables accurate particle tracking that can resolve the exponential stretching of material lines in the flow. Simply extending the Stokes' flow parameters to stirring at finite Reynolds number leads to a decrease in system performance, but tuning the system based on the topological analysis can lead to enhanced stirring.