The curvature of material lines in a three-dimensional chaotic flow

Folding of material filaments was examined computationally in the three-dimensional flow in a cylindrical duct with helical deflectors by tracking the curvature of line elements in the flow. Two geometries were analyzed: a configuration in which the flow is globally chaotic, and an alternative geometry which has a mixture of chaotic and regular motion. The behavior of the curvature field in this complex flow geometry was in agreement with that previously observed for much simpler two-dimensional model flows [Phys. Fluids 8, 75 (1996)]. Curvature profiles along individual element trajectories indicate that an inverse relationship exists between the rates of stretching and curvature. Material elements are compressed when they are folded. After an initial transient, the mean curvature oscillates within a finite range with a periodicity matching that of the flow geometry. The spatial structure of the curvature field becomes period-independent, and the probability density functions of curvature computed for different numbers of periods collapse to an invariant, self-similar distribution without the need for scaling.