  A phase-field method for unstructured grids that is accurate, conservative,
and robust is proposed in this work. The proposed method also results in
bounded transport of volume fraction, and the interface thickness adapts
automatically to local grid size. In addition to this, we present a novel
formulation for two-phase flows on collocated grids that is provably energy
stable, which is a critical feature for robust simulations of two-phase
turbulent flows. The proposed method is first evaluated on canonical test
cases, including scenarios like drop advection and drop in a shear flow. The
accuracy of the proposed method in the presence of grid transitions, which is
important for simulations in complex geometries using unstructured grids, is
also evaluated. We assess the robustness of our method by performing
simulations of a drop in homogeneous isotropic turbulence at infinite Reynolds
number with varying density ratios. Furthermore, as a step toward verification
and validation, simulations of test cases in complex geometries and conditions,
such as, a damped surface wave, infinitesimal interface distortion on a liquid
jet, and the atomization of a liquid jet from the engine combustion network's
Spray A nozzle are presented. These simulations illustrate the accuracy,
robustness, and applicability of the proposed method in various kinds of
complex two-phase flow applications of engineering interest.
