Mixing as a correlated aggregation process

Abstract

Mixing describes the process by which scalars, such as solute concentration or fluid temperature, evolve from an initial heterogeneous state to uniformity under the stirring action of a fluid flow. Mixing occurs initially through the formation of scalar lamellae as a result of fluid stretching and later by their coalescence due to molecular diffusion. Owing to the linearity of the advection-diffusion equation, scalar coalescence can be envisioned as an aggregation process. While random aggregation models have been shown to capture scalar mixing across a range of turbulent flows, we demonstrate here that they are not accurate for most chaotic flows. In particular, we show that the spatial distribution of the number of lamellae in aggregates is highly correlated with their elongation and is also influenced by the fractal geometry that arises from the chaotic flow. The presence of correlations makes mixing less efficient than a completely random aggregation process because lamellae with similar elongations and scalar levels tend to remain isolated from each other. Based on these observations, we propose a correlated aggregation framework that captures the asymptotic mixing dynamics of chaotic flows and predicts the evolution of the scalar pdf based on the flow stretching statistics. We show that correlated aggregation is uniquely determined by a single exponent which quantifies the effective number of random aggregation events, and is dependent on the fractal dimension of the flow. These findings expand aggregation theories to a larger class of systems, which have relevance to various fundamental and applied mixing problems