An operator-theoretic approach to particle advection*

Respecting all of the interactions between a turbulent fluid and individual particle trajectories in a numerical simulation is extremely computationally expensive, since the current state of the system includes contributions from the historical trajectories of each particle. Here we move from a micro-scale mechanistic model of fluid-particle interaction towards a macro-scale statistical model of particle ensembles, while retaining the primary drivers of interaction. We derive a computationally tractable simplification of the Perron-Frobenius-Ruelle integro-differential operator that provides a finite-dimensional projection of the probability density functions of particle concentration. We demonstrate the practical use of our approach in the turbulent breakdown of Taylor-Green vortices, a canonical representation of highly energetic flows. One important feature of our model is that we feed back particle concentrations to influence the fluid phase. Irregular distributions of relatively dense or buoyant ensembles of particles induce a baroclinic torque on the flow, which inevitably couples with the future evolution of the distribution of inertial particles. The influence of the feedback between particle and fluid is shown through analysis of turbulent flow statistics.