# Droplet-turbulence interactions and quasi-equilibrium dynamics in turbulent emulsions

## TL;DR

This paper uses advanced simulations to study how droplets interact with turbulence in emulsions, revealing a quasi-equilibrium cycle of coalescence and breakup, and highlighting the unique flow topologies caused by droplet interfaces.

## Contribution

It demonstrates the capability of the pseudopotential lattice-Boltzmann method for stable, long-term emulsion simulations and uncovers the dynamic limit-cycle behavior of emulsions in turbulence.

## Key findings

- Droplet size distribution follows a d^{-10/3} scaling.
- Emulsions evolve into a quasi-equilibrium cycle of coalescence and breakup.
- Flow topology in emulsions shows dominant vortex compression and axial straining.

## Abstract

We perform direct numerical simulations (DNSs) of emulsions in homogeneous, isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long, stable simulations in certain parameter regions. Varying the dispersed phase volume fraction $\phi$, we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher $\phi$, with the Hinze scale dividing the two effects. Droplet size ($d$) distribution was found to follow the $d^{-10/3}$ scaling (Deane & Stokes 2002). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy $E_k$, enstrophy $\omega^2$ and the droplet number density $N_d$, we find that their dynamics resemble limit-cycles with a time delay. Extreme values in the evolution of $E_k$ manifest in the evolution of $\omega^2$ and $N_d$ with a delay of $\sim0.3\mathcal{T}$ and $\sim0.9\mathcal{T}$ respectively (with $\mathcal{T}$ the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different than single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms become dominant in the droplet phase, a consequence of the elastic behaviour of droplet interfaces. Revised and extended version now published in the Journal of Fluid Mechanics: https://doi.org/10.1017/jfm.2019.654