The unsteady flow evolution in a porous chamber with surface mass injection simulating propellant burning in a nozzleless solid rocket motor has been investigated by means of a large-eddy simulation (LES) technique. Of particular importance is the turbulence-transition mechanism in injection-driven compressible flows with high injection rates in a chamber closed at one end and connected to a divergent nozzle at the exit. The spatially filtered and Favre-averaged conservation equations of mass, momentum and energy are solved for resolved scales. The effect of unresolved subgrid scales is treated by using a dynamic Smagorinsky model extended to compressible flows. Three successive regimes of flow development are observed: laminar, transitional, and fully developed turbulent flow. Surface transpiration facilitates the formation of roller-like vortical structures close to the injection surface. The flow is essentially two-dimensional up to the mid-section of the chamber, with the dominant frequencies of vortex shedding governed by two-dimensional hydrodynamic instability waves. These two-dimensional structures are convected downstream and break into complex three-dimensional eddies. Transition to turbulence occurs further away from the wall than in standard channel flows without mass injection. The peak in turbulence intensity moves closer to the wall in the downstream direction until the surface injection prohibits further penetration of turbulence. The temporal and spatial evolution of the vorticity field obtained herein is significantly different from that of channel flow without transpiration.