Lagrangian diagnostics, such as the finite-time Lyapunov exponent and Lagrangian coherent structures, have become popular tools for analyzing unsteady fluid flows. These diagnostics can help illuminate regions where fluid parcels transported by a flow will converge to and diverge from. Unfortunately calculating Lagrangian diagnostics can be time consuming and computationally expensive. Recently new Eulerian diagnostics, such as objective Eulerian coherent structures and the trajectory divergence rate, have been developed which are faster and less expensive to compute. Because Eulerian diagnostics are so new, there is sill much about their connection to Lagrangian diagnostics that is unknown. This paper will provide a mathematical bridge between Lagrangian and Eulerian diagnostics. It will rigorously explore the deep mathematical relationship that exists between invariants of the Cauchy-Green strain tensor and the Eulerian rate-of-strain tensor in the infinitesimal time limit. Additionally, this paper will develop a new Eulerian diagnostic, infinitesimal-time Lagrangian coherent structures, and show its efficacy in predicting the Lagrangian transport of fluid parcels.
Comments: 30 Pages. In preparation for journal submission
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