  The breakup of a spherical droplet in a decaying homogeneous isotropic
turbulence is studied by solving the Cahn-Hilliard-Navier-Stokes equations,
using the discrete unified gas kinetic scheme combined with the
free-energy-based phase-field model. We focus on the combined effects of
turbulence and surface tension on the breakup process by assuming that the two
fluid phases have the same density and same viscosity. The key physical
parameters of the system include the volume fraction (6.54%), the initial Weber
number (21.7), and the initial Taylor microscale Reynolds number (58). Three
distinct stages of droplet evolution are identified, namely, the deformation
stage when the initially spherical droplet evolves into an irregular geometric
shape with complex structures, the breakup stage when many daughter droplets
are formed, and the restoration stage when the droplets relax towards spherical
shape. These three stages are analyzed systematically from several
perspectives: (1) a geometric perspective concerning the maximum equivalent
diameter, the total number of droplets, total interface area, and probability
distribution of droplet diameters, (2) a dynamic perspective concerning the
evolution of local velocity and vorticity at the fluid-fluid interface, (3) a
global perspective concerning the evolution of average kinetic energy /
dissipation rate and their Fourier spectra, (4) spherical harmonics based
energetics concerning simultaneous transfer of kinetic energy across different
length scales and different radii relative to initial droplet center, and (5)
the time evolution of global kinetic energy and free energy of the system.
