We present a lattice Boltzmann algorithm to solve the Beris-Edwards equations of motion for a cholesteric liquid crystal. We use our algorithm to investigate permeative flow. We find that, for helices pinned at the boundary, a small body force leads to a huge viscosity increase whereas larger ones induce no increase. This shear thinning is in agreement with experiments. If instead, the helix lies perpendicular to the plates, there is almost no viscosity increase. For strong forcing, we identify a flow-induced double twist in the director field. We compare this texture with the static double twist of cubic blue phases.