CDS Thesis Seminar: A Dynamical Systems Approach to Unsteady Systems

Shawn Shadden, Control and Dynamical Systems
California Institute of Technology

For steady systems, interpreting the flow structure is typically straightforward because streamlines and trajectories coincide. Therefore the velocity field, or quantities derived from it, provide a clear description of the flow geometry. For unsteady flows, this is often not the case. A more natural choice is to understand the flow in terms of particle trajectories, i.e. the Lagrangian viewpoint. While the chaotic behavior of trajectories of unsteady systems makes direct interpretation difficult, more structured and frame-independent techniques have been developed. The method presented here uses finite-time Lyapunov exponent (FTLE) fields to locate Lagrangian Coherent Structures (LCS). LCS are codimension one separatrices that partition regions in phase space with dynamically different behavior. This method enables the detection of often non-obvious, time-dependent boundaries in complicated flows which greatly elucidates the transport and mixing geometry. We will discuss the theoretical development of defining LCS from FTLE fields and demonstrate this approach on a range of applications.