Based on the free-water assumption that the water-rich liquid phase contains only pure water, we develop two simple free-water Rachford-Rice methods: 1) a two-phase free-water Rachford-Rice method where the phase fractions can be analytically solved; and 2) a three-phase free-water Rachford-Rice method where there is only one unknown in the objective function (i.e. the vapour-phase fraction) that is used for solving the phase fractions. Combining these two Rachford-Rice methods, a new free-water flash algorithm is developed to perform multiphase flash calculations where single-phase equilibria, two-phase equilibria, and three-phase vapour-liquid-aqueous equilibria can be considered. In this free-water flash algorithm, we first test if the mixture is stable; if the mixture is found to be unstable, we directly initiate the three-phase free-water flash. A set of criteria is developed for one to properly switch from a three-phase free-water flash to either a two-phase free-water flash or a conventional two-phase flash, depending on whether a water-rich phase is present in the two-phase equilibrium. We also develop efficient and robust methods for initializing the equilibrium ratios for the two-phase flashes. The negative flash is allowed in the flash calculation algorithms. A number of example calculations are carried out to demonstrate the robustness of the newly developed algorithm. A good agreement can be achieved between the flash results obtained by the new flash algorithm and those obtained by the conventional full three-phase flash algorithm.