Distributions of Time of Flight and Mixing Efficiency in Three-Dimensional Chaotic Advection

The distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincaré section, also called first-return time in dynamical systems community) are investigated numerically for 3D stationary chaotic mixers. Above all, we study the large tails of those distributions, and show that two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. Hence long time of flight correspond to particles that are stuck for a while in the vicinity of stable manifolds of fixed points. This property is then applied to the rate of decay of scalar energy in the case of diffusing species with global chaos. More precisely, we suggest that mixing in 3D-flows is governed by regions of slow rate of strain rather than by the Lyapunov exponent. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to -3. The model suggests that long times of flights correspond to particles stuck for a while in the vicinity of solid walls.