  Phase-field methods have long been used to model the flow of immiscible
fluids. Their ability to naturally capture interface topological changes is
widely recognized, but their accuracy in simulating flows of real fluids in
practical geometries is not established. We here quantitatively investigate the
convergence of the phase-field method to the sharp-interface limit with
simulations of two-phase pipe flow. We focus on core-annular flows, in which a
highly viscous fluid is lubricated by a less viscous fluid, and validate our
simulations with an analytic laminar solution, a formal linear stability
analysis and also in the fully nonlinear regime. We demonstrate the ability of
the phase-field method to accurately deal with non-rectangular geometry, strong
advection, unsteady fluctuations and large viscosity contrast. We argue that
phase-field methods are very promising for quantitatively studying moderately
turbulent flows, especially at high concentrations of the disperse phase.
