As a conclusion, the study was performed with a asymmetric mesh which induces a little numerical perturbation, and this is this perturbation which makes to disturb flow symmetry.
Although this simulation does not represent, it is very interessant to see flow numerical evolution, function of mesh and of boundaries conditions.

Relaxation coefficients choice is crucial and moreover, according to us, the injector lenght is too short to execute a more time simulation. Indeed, after a lot of iterations vortices make to come back mixture in the room injector.