We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.
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The authors wish to thank Dr Alexander Smits for his input on the results presented in this paper, Mingjun Wei and Milos Ilak for their work with the turbulent channel flow simulation, and Francois Lekien for his helpful thoughts on the Lagrangian stucture analysis. This work was generously supported by an NSF Graduate Research Fellowship, NSF awards CMS-0347239 and DMS-04-04845, and AFOSR grant FA 9550-06-0092.