  Realistic two-phase flow problems of interest often involve high $Re$ flows
with high density ratios. Accurate and robust simulation of such problems
requires special treatments. In this work, we present a consistent,
energy-conserving momentum transport scheme in the context of a second order
mass-conserving phase field method. This is achieved by (1) accounting for the
mass flux associated with the right-hand-side of the phase field equation in
the convective flux of the conservative form of the momentum transport
equation---a correction absent in previous phase field simulations (2)
utilization of non-dissipative spatial discretization. We demonstrate accuracy
and robustness improvements from our proposed scheme via numerical tests,
including a turbulent case of a water jet subject to air cross-flow. Our
proposed modifications to the momentum transport equation can be quite readily
extended to general conservative phase field methods. Additionally, in the
framework of this phase field method, we present a free energy-based surface
tension force calculation scheme. This scheme, which significantly reduces
spurious currents, is based on a general paradigm that can also be extended to
other two-phase flow methods.
