Detection of Lagrangian coherent structures in three-dimensional turbulence

Abstract

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.

Pinch-off of non-axisymmetric vortex rings

“…LCS are finitetime analogues to the stable and unstable manifolds, for systems with arbitrary time dependence. LCS were introduced in a series of papers by Haller and coworkers (Haller & Poje 1998;Haller 2000Haller , 2001aHaller ,b, 2002Haller & Yuan 2000), and have been employed for structure identification and for investigating vortex dynamics by Shadden, Dabiri & Marsden (2006) and Shadden et al (2007), Green, Rowley & Haller (2007) and Green, Rowley & Smits (2010), O'Farrell & Dabiri (2010) and others.…”

“…Hence, a Galilean transformation to a suitable frame of reference could not be easily performed, and the aforementioned vortex bubble methods were not well suited to these complex flows (O'Farrell & Dabiri 2010). In addition, the LCS method has the advantage of being robust to errors in the velocity field, including those arising from interpolation of measured velocity data (Haller 2002), thus allowing the calculation of FTLE fields of much higher resolution than the original data (Green et al 2007).…”

“…This is especially true when the perceived size and shape of the vortices varies depending on the thresholds or contour levels in use. In contrast, LCS often prove useful in these cases because they are frame invariant, their location is independent of threshold selection, and they identify kinematically distinct regions even in complex vortex flows (Shadden et al 2006;Green et al 2007;O'Farrell & Dabiri 2010). Therefore, to resolve the ambiguity in defining the boundary of the leading vortex ring, we considered the LCS in the flow.…”

Coherent structures and the saturation of a nonlinear dynamo

“…1. The remainder of this paper focuses on the backward-time maximum FTLE field, since they reveal the attracting LCSs, which correspond to structures seen using flow visualization in experiments (Green et al 2007). Figure 8 shows the backward-time maximum FTLE field computed for τ corresponding to 9 turnover time units at t 0 = 100 (left) and t 0 = 1700 (right) from a grid of initial conditions with 512 × 512 particles.…”

“…Attracting LCSs have commonly been associated with local maximizing curves (ridges) in the backward-time finite-time Lyapunov exponent (FTLE) field and repelling LCSs to ridges in the forward-time FTLE field (Shadden et al 2005;Green et al 2007;BeronVera & Olascoaga 2010). There are limitations in such definition, as pointed out by Haller (2011) andFarazmand &Haller (2012), e. g., a ridge in the FTLE field may indicate the presence of a shear LCS or no LCS at all.…”