Integrated computation of Lagrangian coherent structures during DNS of unsteady and turbulent flows

The computation of Lagrangian coherent structures (LCS) typically involves post processing of experimentally or numerically obtained fluid velocity fields to obtain the finite time Lyapunov exponent (FTLE) via a sequence of flow maps (vector fields which describe fluid displacement patterns over a finite time interval, $t_0 \pm T$). However, this procedure can be prohibitively expensive for large-scale complex flows of engineering interest. In this work, an alternative approach involving computation of the FTLE on the fly during direct numerical simulation (DNS) of the 3D Navier-Stokes equations is developed. This incorporation of the FTLE computations into a parallel DNS solver relies on Lagrangian particle tracking to compose forward time flow maps, and an Eulerian treatment of the backward time flow map [Leung, J. Comp. Physics 2011] coupled with a semi-Lagrangian advection scheme. The time $T$ flow maps are accurately constructed from smaller sub-steps [Brunton \& Rowley, Chaos 2010], resulting in low CPU and memory requirements for computing evolving FTLE fields. Illustrative examples will be presented to demonstrate the capability of the approach including the evolution of a turbulent vortex ring and turbulent flows in complex porous media.