Studying how small inertial particles suspended in turbulent flows move relative to each other provides fundamental insights into their transport, mixing and collisions. These insights are crucial for tackling diverse problems ranging from droplet growth in warm clouds, to planetesimal formation through collisional aggregation in turbulent protoplanetary nebula. A deeper understanding of the relative motion of the particles can be obtained by investigating both their forward-in-time (FIT) and backward-in-time (BIT) dispersion. When FIT and BIT dispersion are different it signifies irreversibility, and since FIT and BIT dispersion are related to different problems, understanding the irreversibility is of fundamental and practical importance.
I will present new theoretical arguments and asymptotic predictions, along with results from Direct Numerical Simulations (DNS) of the governing equations, to show that inertial particle dispersion can be very strongly irreversible in turbulence, with BIT being much faster than FIT dispersion in 3-dimensional turbulence. The results also show that inertial particles can disperse much faster than fluid (interialess) particles. I will also present arguments, confirmed by DNS results, that in 2-dimensional turbulence the nature of the irreversibility and the direction of the particle energy fluxes can invert when the particle inertia exceeds a certain threshold. These results significantly advance our understanding of dispersion problems, and lead to new capabilities for predicting the effect of inertia on the rate at which particles spread out and mix together in turbulence, and the rate at which they collide.

The ability to effectively control a fluid would enable many exciting technological advances, such as the design of quieter, more efficient aircraft. Model-based feedback control is a particularly attractive approach, but the equations governing the fluid, although known, are typically too complex to apply standard tools for dynamical systems analysis or control synthesis. This talk addresses model reduction techniques, which are used to simplify existing models, to obtain low-order models tractable enough to be used for analysis and control, while retaining the essential physics. In particular, we will discuss two techniques: balanced truncation and Koopman modes. Balanced truncation is a well-known technique for model reduction of linear systems, with provable error bounds, but it is not computationally tractable for very large systems. We present an approximate version, called Balanced POD, that is computationally tractable, and produces much better models than traditional Proper Orthogonal Decomposition (POD), at least for the examples studied. Koopman modes are based on spectral analysis of the Koopman operator, an infinite-dimensional linear operator that describes the full nonlinear dynamics of a nonlinear system, and we show how the associated modes can elucidate coherent structures in examples including a jet in crossflow and the wake of a flat plate.