Topological chaos in fluids:
braids, ghost rods, and almost invariant sets
Mark Stremler
Department of Engineering Science & Mechanics
Virginia Tech, USA
Abstract:
Mixing, blending, and separation of fluid-based substances are related activities that are heavily influenced by the underlying fluid motion. In laminar flows, the phenomenon of `chaotic advection', in which passive particles carried by a periodic laminar flow experience chaotic trajectories, has proven to be a powerful tool. However, many of the traditional dynamical systems techniques for analyzing chaotic advection rely on detailed understanding of the fluid dynamics throughout the entire domain of interest. This dependence on the dynamics, and the corresponding importance of system perturbations, makes it difficult in many cases to extrapolate results to practical systems with complex fluids and/or incomplete data. Developments over the past decade in `topological chaos' show promise for making this extrapolation possible. I will discuss topological chaos in the context of fluid mixing, including several of the recent developments.
In two-dimensional time-dependent flows or three-dimensional flows with a certain symmetry, the relative motion of periodic orbits provides a framework for analyzing chaos in the system through application of the Thurston-Nielsen (TN) classification theorem. This theorem establishes a quantitative lower bound on the complexity of the flow simply by examining the topology, or braiding, of the periodic orbits. These orbits can correspond to imposed motion of solid internal boundaries or to fluid particle trajectories generated by the dynamics of the flow. This approach makes it possible to predict the presence of chaotic advection in a stirred system using only a limited amount of data.
`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. In some systems these ghost rods are clearly visible as elliptic islands in a Poincare section, but often the key periodic orbits can be difficult to identify. Furthermore, there appear to be situations in which perturbations remove periodic orbits from the system, but characteristics of the topological chaos still persist. I will 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 new approach for extending the use of the TN classification theorem to a variety of fluid systems.