The flow of granular matter is highly influenced by the properties of the constitutive grains. Among these properties, the shape is most likely one of the more influential. This is even more critical when the grain-shape is getting highly concave (e.g., star-shaped) since the packing properties exhibit non-trivial features like high porosity level and sharp jamming/unjamming transition. Despite the richness and ubiquitousness of these systems, very few is known about their flow behavior and the local mechanisms involved. By means of 2D Contact Dynamics simulations, we investigate the flow properties of strongly concave grains within a rotating drum. The system has the advantage of presenting, in a single test, almost all the flow characteristics that have been observed in granular systems: a solid to quasi-static and inertial flow phases. The shape of the grains is systematically varied from disks to crosses made of four very thin arms. Rotation speeds are also varied, while making sure to remain in a dense and continuous flow regime with a well-defined free surface for all shapes. In general terms we find that, for a given rotation speed, the slope of the free surface increases with the concavity and then saturates at a certain grain concavity value. The stress profiles vary significantly with grain shape, but still exhibit common generic trends. For example, the shear stress profile is characterized by an initial linear increase as the depth increases, then becomes independent of the depth close the drum walls. On the contrary, the normal stresses decrease linearly with depth. Momentum balance equations allows to correctly predict the normal stress profile for all shapes and drum velocities but fails in predicting the shear stress profiles. By postulating that in the drum geometry annular regions of small thicknesses (analogous to the Couette shear) exist, we develop a new model that perfectly predict the shear stress profiles in a drum, whatever the shape of the grains and the rotational speeds