A numerical method for the solution to the density-dependent incompressible Navier-Stokes equationsmodeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows tomodel the flow of an arbitrary number of liquid phases together with an additional vacuum phase separatedwith a free surface. It is based on a volume-of-fluid (VOF) approach involving N indicator functions (one perphase, identified by its density) that guarantees mass conservation within each phase. An additional indicatorfunction for the whole liquid domain allows to treat boundary conditions at the interface between the liquiddomain and a vacuum. The system of partial differential equations is solved by implicit operator splitting ateach time step: first, transport equations are solved by a forward characteristics method on a fine Cartesiangrid to predict the new location of each liquid phase ; second, a generalized Stokes problem with a density-dependent viscosity is solved with a finite element method on a coarser mesh of the liquid domain. A novelalgorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the Nphases is validated on benchmark flows. Then, we focus on a novel application, and compare the numericaland physical simulations of impulse waves, i.e. waves generated at the free surface of a water basin initiallyat rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is topredict the effects of a landslide-generated impulse wave in a reservoir.