This study presents the implementation and validation of a second-order accurate solver for the 4-equation multi-fluid method in a cell-vertex context, to handle aeronautical air-assisted liquid sheet configurations. Validations include one and two-dimensional shock tube problems, Laplace tests, droplets oscillations, and a two-dimensional configuration reminiscent of an aeronautical airblast injector. Promising results are obtained in the last case, especially when the pressure is increased and the characteristic flow times decreased, a trend observed in recent aeronautical developments, but not necessary in classical academical benchmarks. Being fully compatible of the reactive flow formulation of the cell-vertex solver AVBP, this study paves the way to future monolithic simulations of airblast injectors for aeronautical combustion chambers, including both the multi-phase and the reactive regions.