Optimal initial condition of passive tracers for their maximal mixing in finite time

摘要

The efficiency of a fluid mixing device is often limited by fundamental lawsand/or design constraints, such that a perfectly homogeneous mixture cannot beobtained in finite time. Here, we address the natural corollary question: Giventhe best available mixer, what is the optimal initial tracer pattern that leadsto the most homogeneous mixture after a prescribed finite time? For idealpassive tracers, we show that this optimal initial condition coincides with theright singular vector (corresponding to the smallest singular value) of asuitably truncated Perron-Frobenius (PF) operator. The truncation of the PFoperator is made under the assumption that there is a small length-scalethreshold $\ell_\nu$ under which the tracer blobs are considered, for allpractical purposes, completely mixed. We demonstrate our results on twoexamples: a prototypical model known as the sine flow and a direct numericalsimulation of two-dimensional turbulence. Evaluating the optimal initialcondition through this framework only requires the position of a dense grid offluid particles at the final instance and their preimages at the initialinstance of the prescribed time interval. As such, our framework can be readilyapplied to flows where such data is available through numerical simulations orexperimental measurements.