Two different approaches for numerical simulations of primary break-up of liquid jet are described. In the ELSA model (Eulerian Lagrangian Spray Atomization) the liquid and the gas are considered as two species of a same mixture, but without necessarily the same velocity. The drift velocity and the liquid mass dispersion are exactly related to the turbulent liquid mass flux. To describe the topology of the spray the liquid/gas interface density per unit of volume is used instead of the droplet diameter that has no meaning in the dense part of the spray. To establish definitely the model for the turbulent liquid mass flux and for the liquid/gas interface density equation, DNS simulations are carried out to obtain information in the dense zone of the spray where nearly no experimental data are available. Interface tracking is ensured by the Level Set Method and the Ghost Fluid Method (GFM) is used to capture accurately sharp discontinuities for pressure, density and viscosity. Coupling between the Level Set and VOF method is used for mass conservation. Fluid motion is predicted with a projection method for incompressible flows.