This study focuses on a water injection system, typical of those used to inject water in the motor jet plume of rocket engines to reduce noise, and particularly on the destabilization of the liquid jet that leads to a spray system. For that purpose, a large-eddy simulation is conducted for an experimental water jet of the literature. It is a sprinkler whose Reynolds number is and nozzle exit diameter is , similar to injectors used for jet noise reduction studies at a reduced scale. A diffuse interface method is used to calculate the dense liquid phase, and the dispersed phase containing droplets is calculated with an Eulerian solver. Transfers between the dense and the dispersed phases are modeled in the coupling procedure, including the atomization and impingement processes. The jet mean radius and initial instability frequency are consistent with the experimental data, while the position of the jet transition varies with the fluctuation levels at the nozzle exit. The droplets are created in the shear layer and accumulate on the jet center when only the atomization process is considered. When also implemented, the impingement phenomenon, however, is shown to absorb the droplets before they reach the center.