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Dual-mode scramjet combustor is the crucial component for the combined cycle engine of space mission vehicles. The Reynolds Averaged Navier-Stokes (RANS) equations, Spalart-Allmaras turbulence model and the finite-rate reaction model have been utilized to investigate the mixing and combustion in a dual-mode combustor with backward-facing steps, and the species distributions and the Mach number profile of a turbulent diffusion combustion problem have been employed to validate the numerical approach. Moreover, the influences of the injection strategy and the fuel equivalence ratio arrangement on its mixing and combustion flow fields have been explored. The obtained results show that the vertical injection is beneficial for mode transition, and an obvious high pressure region is generated with the vertical injection strategy. Accordingly, its mass-weighted average Mach number drops more sharply. The lower total equivalence ratio is beneficial for the mixing augmentation, as well the smaller fuel equivalence ratio discrepancy. However, the larger total equivalence ratio is beneficial for the mode transition.