STRETCHING AND BENDING OF LINE ELEMENTS IN RANDOM FLOWS

We study the stretching and bending of line elements transported in ra ndom flows with known Eulerian statistics in two and three dimensions. By making use of a cumulant expansion for the log-size of material el ements we are able to analyse the exponential stretching they exhibit in random flows and identify conditions under which it will and will n ot occur. The results are confirmed by our numerical simulation. We al so examine the. evolution of curvature in material elements and confir m by numerical simulation that it is governed by an appropriate versio n of the Pope equation. By modelling this equation as stochastic diffe rential equation we are able to explain the appearance of a power-law tail in the probability distribution for large curvature observed by P ope, Yeung & Girimaji (1989) for surface elements. In two dimensions t he appearance of the tail can indeed be attributed to the occurrence o f events in which the material element undergoes contraction rather th an stretching while subject to bending. In three dimensions the relati onship between episodes of contraction and strong bending is less dire ct. 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 .