An analysis is given, based on rapid distortion theory, of the effect on homogeneous turbulence when it passes through a linear cascade of thin flat plates or aerofoils and in particular the turbulence that emerges on the downstream side. The main results are evaluated for the limiting case in which the spacing ratio of the cascade is very small so that the cascade tends to completely suppress the normal component of velocity of the turbulence everywhere, both within the cascade and downstream of it, resulting in a ‘planar’ (quasi-two-dimensional) turbulence. Some comparisons with experimental measurements of grid generated turbulence passed through cascades of thin aerofoils and in the regions adjacent to the wakes of isolated streamwise plates in a wind tunnel are shown.