The first half of this paper reviews mixing in chaotic flows. The sine-flow is employed as a two dimensional example to demonstrate techniques used for characterizing mixing behavior. Several manifestations of self-similarity are readily apparent. The results of tracer mixing simulations demonstrate a self-similar, iterative development of partially mixed structures in the flow. The spatial distribution of mixing intensities present in the flow is examined via computation of stretching. The probability density function (PDF) of the logarithm of stretching values reveals a Gaussian distribution over the central spectrum of stretching intensities for the globally chaotic case, but contains two peaks for a case with coexisting chaotic and regular regions: a broad Gaussian peak for higher stretching values, corresponding to the chaotic region, and a sharp peak of low stretching values corresponding to the regular regions. The self-similar stretching distributions can be collapsed to a single invariant distribution using an appropriate scaling based on the central limit theorem. The folding processes in the flow are examined through curvature calculations; PDFs for curvature collapse to time invariant self-similar distributions without the need for scaling. Direct computation of the striation thickness distribution (STD) provides the most fundamental (and computationally most expensive) measure of mixing; STDs develop a self-similar form that can be collapsed to an invariant distribution using a simple scaling. The second half of the paper focuses on a real, three-dimensional mixing system: the Kenics static mixer. Two alternate configurations of the Kenics mixer were analyzed: one in which elements have alternating right-handed and left-handed twist (R-L) and a second in which all elements have right-handed twist (R-R). Poincaré sections as well as experiments indicate that the R-L configuration is globally chaotic, while the R-R configuration contains significant segregated, regular regions. Stretching histories of material elements in the two flows were computed, once again revealing self-similar distributions that can be collapsed to an invariant limit using a scaling based on the central limit theorem.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Geometry and Topology
- Modeling and Simulation