Liquid fuel injection modeling and simulation in automotive engines face new challenges related to the need for predictive simulations of combustion regimes. Recently, derived from a statistical approach at mesoscopic level, a high order method of moments coupled to realizable, robust and accurate numerical methods [1], has been obtained and shown to describe properly the dynamics of polydisperse evaporating sprays and its coupling to a turbulent gaseous flow field [2]. However, building up a global multi-scale model with the capability to resolve the whole injection process requires a major breakthrough in terms of both modeling and numerical methods. A new model for evaporating polydisperse sprays with easy coupling capabilities to the separated phases zone is proposed in the present contribution, as well as specific numerical methods and implementation in the p4est library [3] for adaptive mesh refinement and massively parallel computing. The key ingredient is a good choice of variables, which can describe both the polydisperse character of a spray as well as the topology of an interface. After verification cases, some two-phase simulations, challenging both in terms of physics and of high performance computing, have been conducted using adaptive mesh refinement [4]. We focus here on a 3D simulation of a spray in the presence of a frozen Homogeneous Isotropic Turbulence (HIT) gaseous carrier flow and assess the ability of the model and of the related numerical methods to capture the physics of such flows.