Computational fluid dynamics (CFD) has the potential to improve the historical rocket injector design process by simulating the sensitivity of performance and injector-driven thermal environments to the details of the injector geometry and key operational parameters. Methodical verification and validation efforts on a range of coaxial injector elements have shown the current production CFD capability must be improved in order to quantitatively impact the injector design process. This paper documents the status of an effort to understand and compare the predictive capabilities and resource requirements of a range of CFD methodologies on a set of injector model problems. Preliminary results from a steady Reynolds-Average Navier-Stokes (RANS), an unsteady Reynolds-Average Navier-Stokes (URANS) and three different Large Eddy Simulation (LES) techniques used to model a single element coaxial injector using gaseous oxygen and gaseous hydrogen propellants are presented. Initial observations are made comparing instantaneous results and corresponding time-averaged results with steady-state solutions in the near-injector flow field. Significant differences in the flow fields exist, as expected, and are discussed. An important preliminary result is the identification of a fundamental mixing mechanism, accounted for by URANS and LES, but missing in the steady RANS methodology. Since propellant mixing is the core injector function, this mixing process may prove to have a profound effect on the ability to more correctly simulate injector performance and resulting thermal environments. Issues important to unifying the basis for future comparison such as solution initialization, required run time and grid resolution are addressed.