Stretching and bending of line elements in random flows

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Abstract
We study the stretching and bending of line elements transported in random flows with known Eulerian statistics in two and three dimensions. By making use of a cumulant expansion for the log-size of material elements we are able to analyse the exponential stretching they exhibit in random flows and identify conditions under which it will and will not occur. The results are confirmed by our numerical simulation. We also examine the evolution of curvature in material elements and confirm by numerical simulation that it is governed by an appropriate version of the Pope equation. By modelling this equation as stochastic differential equation we are able to explain the appearance of a power-law tail in the probability distribution for large curvature observed by Pope, Yeung & Girimaji (1989) for surface elements. In two dimensions the appearance of the tail can indeed be attributed to the occurrence of events in which the material element undergoes contraction rather than stretching while subject to bending. In three dimensions the relationship between episodes of contraction and strong bending is less direct. This power-law tail allows us to reconcile the observed asymptotic stability, which we confirm here, of the powers and cumulants of the log-curvature with the unboundedness of powers of the curvature itself.

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