The numerical simulation of fluid dynamics and combustion in cryogenic rocket engines is addressed in this paper, with the intent to elucidate flame stabilization mechanisms. A model configuration is devised to allow a fully resolved simulation, both for the dynamics and the flame structure: a two-dimensional splitter plate represents the lip of an injector and the operating point is typical of a real engine. The non-reacting flow field is first scrutinized to evaluate the impact of the large density gradients between the fuel (hydrogen) and oxidizer (oxygen) streams. It is found that the turbulence generated by the splitter is very intense and strongly distorts the high-density-gradient front at both small and large scales. Under reacting conditions, the flame stabilizes right at the lip of the injector, which is a common feature of hydrogen / oxygen flames under these conditions. A particularly complex flame structure is evidenced at the anchoring point, with turbulent transport playing an important role.