In the field of liquid rocket propulsion, ignition, propagation and stabilization of the flame are of first importance for the design of the engine. Computational fluid dynamics (CFD) solvers may provide a great deal of help to proceed with the primary design choice but need to be fed with suited physical models. Important modelling efforts are therefore required to provide reliable computational representations able to take into account compressibility effects, turbulent mixing and chemical kinetics in two-phase flows since ergols are injected at the liquid state. A Lagrangian model has been implemented in the compressible solver N3S-Natur so as to obtain a computational tool able to compute the transient ignition of rocket engines. The physical processes involved at each step of this ignition sequence have been integrated and validated on academically configurations. Three significant contributions rose from this work. First of all, it is highlighted that the description of the micro-mixing is of first importance to correctly capture the flame development. This study also emphasized the need to consider the transient heating of liquid oxygen droplets in order to accurately compute the flame stabilization. Finally, the notion of mixture fraction must be extended to practical devices implying more than two inlets. The proposed approach which is based on the introduction of a fictive injector is not only well suited to rocket engine ignition application but also to deal with other practical devices implying two inlets and more.