  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
