Study of the dynamics of transport and mixing using set oriented methods

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Virginia Tech

Efficient mixing can be achieved in flows where turbulence is absent, if the trajectories of passively advected particles in the flow are chaotic. The chaotic nature of particle trajectories results in exponential stretching of material lines in the flow. Thus the interface along which diffusion occurs is stretched exponentially leading to efficient mixing. It has been demonstrated recently that regions in flow fields that exhibit poor mixing and non-chaotic particle trajectories can have an important bearing on the overall dynamics and transport of the entire domain.

The space-time trajectories of physical stirrers or elliptic points in two dimensional flows can be classified according to braid groups. One can predict a lower bound on the topological entropy (i.e. exponential rate of stretching of material lines) of flows (hf) by applying the Thurston-Nielsen classification theorems to these braids. This gives a reduced order model for the dynamics of transport of the entire flow field using just a few points. Recent work has shown that this methodology can be used to estimate a lower bound on hf using the braids formed by Almost Cyclic Sets (ACS) in certain periodic Stokes' flows. These ACS are closely related to Almost Invariant Sets (AIS) which are identified using a probabilistic set oriented method that makes use of the descritised Perron-Frobenius operator of the flow map.

This work extends this approach to flows at non-zero Reynolds numbers, which take into account the effects of inertia. The role of Finite Time Coherent Structures (FTCS) in the dynamics of flow fields is also investigated. Unlike ACS, the FTCS approach is more general as it can be applied to aperiodic flow fields. Further, the relationship between mixing efficiency and the topological entropy of flow fields at non-zero Reynolds numbers is also studied.

Chaotic advection, mixing, coherent sets