To investigate the flame stability in a solid-fuel ramjet combustor, time-accurate calculations using a compressible flow solver with a modified Godunov flux-splitting scheme have been performed on high Reynolds number turbulent non-premixed reacting flows over a backward-facing step with mass bleed on one wall. The combustion process considered was a one-step, irreversible, and finite rate chemical reaction. The numerical results for reacting flows show that the two-dimensional (2-D) simulations can provide reasonable predictions on the dimensionless particle number decay rate and residence time in the flame holding recirculation zone, evolutions of both axial and transverse mean velocity profiles, and critical characteristic exhaust velocity separating the sustained combustion from the non-sustained combustion. In addition to the validation of 2-D reacting flow calculations, two- and three-dimensionally computed mean-velocity profiles are compared with existing experimental data for isothermal flows to check the suitability of 2-D simulations on capturing the large-scale mean flows.