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 co-dimension 1 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.The first portion of this thesis deals with the theoretical development of LCS for two-, and then, n-dimensional systems, where n>2. Based on the definitions presented, some important properties of these structures are proven. It is shown that the flux across an LCS is typically very small and depends on the relative strength of the structure, the difference between the local rotation rate of the LCS with that of the Eulerian velocity field, and the integration time used to compute the FTLE field.The second portion of the thesis presents a series of numerical studies in which LCS are used to examine a range of interesting applications. This portion is bridged with the theoretical development presented in the first half by a brief chapter describing the numerical computation of FTLE fields and LCS. Applications presented in the second half of the thesis include the study of vortex rings in which LCS are used to define the unsteady vortex boundary to clarify the entrainment and detrainment processes; the computation of LCS in the ocean to provide mesoscale separatrices that help characterize the flow conditions and help navigate gliders or drifters used for sampling; flow over an airfoil where an LCS captures the unsteady separation profile; flow through a micro-mixing channel where LCS reveal the mechanism and geometry of chaotic mixing.
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