A detailed understanding of the driving mechanisms behind primary atomization is crucial to the optimization of sprays for efficient combustion in modern propulsion systems. Many challenges are associated with simulating realistic turbulent atomization, such as the multiplicity of length and time scales of the turbulent flow field and gas-liquid interface, discontinuous fluid properties and pressure at the phase interface, high density ratios that degrade numerical robustness, and complex shapes of spray injectors. These challenges have hindered progress in computational modeling of atomizing two-phase flows, and as a result a complete characterization of all physical processes involved in turbulent atomization has remained elusive. This paper presents a suite of computational tools that have been developed in an effort to simulate primary atomization from first principles. The incompressible Navier-Stokes equations are handled in the context of a high-order accurate, discretely conservative, finite difference solver shown to be ideally suited for direct numerical and large-eddy simulations of turbulence. A conservative level set method is used for interface capture, improved through the use of local re-initialization enabled by an efficient fast marching method. A high-density ratio correction algorithm is employed that leads to tighter coupling between mass and momentum transport. Finally, the use of immersed boundaries allows for modeling of complex geometries without requiring body-fitted meshes, eliminating time spent generating complex grids. The framework outlined herein is shown to have the ability to capture important instabilities for atomizing flows, such as Rayleigh-Plateau and Kelvin-Helmholtz instabilities. Simulations of air-assisted breakup of both planar and coaxial liquid layers are shown to agree well with theoretical and experimental results. This strategy is employed to simulate the breakup of a turbulent liquid jet under diesel conditions, the atomization of a liquid sheet issued from a pressure swirl atomizer, and finally a complete dual-orifice atomizer, leading to qualitative insights on the atomization process. Detailed parallel scaling results are also provided.