We use lattice Boltzmann simulations to solve the Beris-Edwards equations of motion for a cholesteric liquid crystal subjected to Poiseuille flow along the direction of the helical axis (permeative flow). The results allow us to clarify and extend the approximate analytic treatments currently available. We find that if the cholesteric helix is pinned at the boundaries there is an enormous viscosity increase. If, instead, the helix is free the velocity profile is flattened, but the viscosity is essentially unchanged. We highlight the importance of secondary flows, and, for higher flow velocities, we identify a flow-induced double twist structure in the director field--reminiscent of the texture characteristic of blue phases.