  A two-phase, low-Mach-number flow solver is created and verified for
variable-density liquid and gas with phase change. The interface is sharply
captured using a split Volume-of-Fluid method generalized for a
non-divergence-free liquid velocity and with mass exchange across the
interface. Mass conservation to machine-error precision is achieved in the
limit of incompressible liquid. This model is implemented for two-phase
mixtures at supercritical pressure but subcritical temperature conditions for
the liquid, as it is common in the early times of liquid hydrocarbon injection
under real-engine conditions. The dissolution of the gas species into the
liquid phase is enhanced, and vaporization or condensation can occur
simultaneously at different interface locations. Greater numerical challenges
appear compared to incompressible two-phase solvers that are successfully
addressed for the first time: (a) local thermodynamic phase equilibrium (LTE)
and jump conditions determine the interface solution (e.g., temperature,
composition, surface-tension coefficient); (b) a real-fluid thermodynamic model
is considered; and (c) phase-wise values for certain variables (e.g., velocity)
are obtained via extrapolation techniques. The increased numerical cost is
alleviated with a split pressure-gradient technique to solve the pressure
Poisson equation (PPE) for the low-Mach-number flow. Thus, a Fast Fourier
Transform (FFT) method is implemented, directly solving the continuity
constraint without an iterative process. Various verification tests show the
accuracy and viability of the current approach. Then, the growth of surface
instabilities in a binary system composed of liquid n-decane and gaseous oxygen
at supercritical pressures for n-decane is analyzed. Other features of
supercritical liquid injection are also shown.
