Previous numerical simulations of the hydrogen-air rotating detonation engine have been carried out using a simple fuel-injection pattern. This pattern assumes that the fresh gas mixture is injected through every point of the head-wall. This article improves the simulation by discussing four new injection patterns. Parts of the wall are treated as a solid wall in the new patterns. The simulation results show that slits or intervals on the head-walls cause interactions between the detonation waves and combustion products, which affect the formation and stability of detonation. Unstable detonation should be a normal state of rotating detonation engines between stable detonation and failed detonation. The features of the flow fields can be used as a reference for combustion chamber design.